Even if you are a creationist, it's important to understand not only what other people believe, but also why they find it compelling. If you cannot fathom how any sane, rational person could believe in evolution, then maybe what you call 'evolution' is not actually what people believe. Maybe they believe something else, instead.
So let's take a look at some lines of evidence...
Fact: Fossils are found in layers of sedimentary rock, but not all fossils are found in all layers. Deeper layers generally indicate an earlier time period and vice versa. Less complex fossils are always found in earlier layers, while more complex fossils are always found in more recent layers. This is the same for fossils of plants as well as animals. Most of the fossils in earlier layers belong to species that are now extinct.
Fossils are a fascinating subject all of their own. Paleontologists study fossils to learn about what the world looked like in the past. We understand how fossils are formed, and we can determine a great deal of information from not only the fossils themselves, but also where they were located.
One of the most striking facts about the fossil record is in the way fossils are layered in sedimentary rock. Due to the way these sediments were laid down, we can conclude that, generally speaking, lower layers formed before higher ones. The study of these geological layers is called Stratigraphy.
What we find everywhere in the fossil record is that the more recent layers contain fossils of more complex life forms, while older layers contain fossils of less complex life forms. This presents an immediate puzzle. What does this tell us about the history of life on earth?
The most natural explanation is that earlier life forms were indeed less complex and that living organisms became more complex over time. Not individually of course, but rather successive generations of organisms became gradually more complex than the last. We can also work out roughly when each sedimentary layer was formed, and thus calculate a rough timeline of the development (or, 'evolution') of life on earth. When scientists talk about fossils being millions or billions of years old, these are not wild guesses. They are actual measurements using very precise techniques. More on that later.
For creationists, however, the difficulty is clear. At a minimum, they must reject any notion of a long timeline of events (depending on their view of creation). Some suggest that the layers were all laid down during Noah's flood. They suggest that the layers were formed through cataclysmic events, and that the reason for the layering of fossils from least to most complex is due to the more complex creatures being more able to reach higher ground during the early stages of the flood.
The shortcomings of this answer should be fairly obvious, but I'll list just two major flaws. Perhaps you can find more. Firstly, studies of the Grand Canyon reveal nearly 40 layers, including signs of erosion between some layers. This can only be explained by many separate geological events, over a very long period of time, with additional long periods in between. Secondly, it's simply absurd to think that dinosaurs (including ones that could fly) failed to beat sloths to higher ground. But in case you do think it's plausible, I'll leave you to ponder how all of the plants and trees managed to pull off the very same feat. Then there's the fact that fossils of Kangaroos are only found in Australia...
Fact: Despite what creationists may tell you, Radiometric Dating methods, including Carbon Dating, are reliable. We know this because they have been tested extensively on samples of known age, and also compared against other methods such as ice cores and tree rings. Not only do scientists know they work, they understand how they work, and the limitations of each method.
Fact: Most fossils are not dated using Carbon Dating. This is because Carbon Dating is only reliable for biological organisms up to 50,000 years old. It is not used on rocks, and organisms older than 50,000 years old, because they don't contain any (original) carbon.
One common tactic of creationists is to insist that while we have a huge array of evidence from the fossil record, we have no way of knowing how old each fossil really is. This is quite simply dishonest. You've probably already heard of Carbon Dating and if you're a creationist, you probably already "know" that it's not reliable. But if so I'd like to ask you to reconsider, and I will present what I think is some pretty compelling evidence.
The first thing to point out is that, despite what you may have heard, Carbon Dating has been rigorously tested, including against samples of known age. Scientists are not stupid. Further, in almost all cases, several samples are collected from different parts of an organism. If Carbon Dating produced wildly inaccurate or random results, it would be difficult to explain how the results from multiple samples agree. Most (if not all) of the creationist claims about unreliable results from Carbon Dating are a result of contamination or creationists who didn't know what they were doing.
The second thing to point out is that Carbon Dating only works for samples up to around 50,000 years old. How do we know that? Well, because again scientists test it extensively to determine how accurate it is, and where it can or cannot be used. They also understand how and why it works, and they can predict how long it takes for the original Carbon-14 in an organism to decay.
What does this mean? Well, for the majority of fossils, Carbon Dating isn't used. That's because for fossils that are millions of years old, there's none of the original carbon left in them. But how do we know they are millions of years old? Well, we can determine when the surrounding sedimentary rock layer was laid down.
One method is Radiometric Dating, which works similar to Carbon Dating but using other isotopes that decay more slowly. Has it been thoroughly tested? Yes.
Is it reliable? Consider this. Radiometric Dating can be used with several different isotopes, all with different decay rates. Yet all of the methods agree, not only with each other but also with other methods as appropriate, including ice cores, and tree rings.
Some Christians have argued that something may be slowly changing with time so all the ages look older than they really are. The only two quantities in the exponent of a decay rate equation are the half-life and the time. So for ages to appear longer than actual, all the half-lives would have to be changing in sync with each other.
Radiometric Dating: A Christian Perspective
Have radioactive isotopes been changing? No, they haven't.
The simple fact is that scientists don't just cherry-pick data and use random methods to arrive at some desired conclusion. That's just not how science works. The various dating methods do rely on some assumptions, but even those assumptions can be (and are) tested. We know the methods work, and we also know their limitations. I'd encourage you to research this directly from the scientific sources, rather than creationist websites. By all means read the creationist websites as well, but check all of their claims against the data. You'll even find plenty of responses to creationist claims within the scientific literature.
By the way, the age of the Dead Sea Scrolls was determined via Carbon Dating.
Fact: Humans share at least 95% of our DNA with chimpanzees.
Fact: Some viruses insert their DNA at random positions in the genome of the host, which, if it affects the germ cells, can be passed on to descendants. There are several instances of such viral insertions in exactly the same locations in both the human and chimpanzee genomes.
Fact: By comparing the DNA of various animals and using markers such as insertions of viral DNA, scientists can plot the family tree of these animals. When they do this, it matches almost exactly with the relationship structure that had already been determined via comparative morphology. Such a match is difficult to explain if all animals do not share common ancestry.
The evidence just from fossils is probably enough on its own, but there is another significant piece of evidence that points strongly to evolution. Our DNA. Our genome consists of genes, which are contained in long strands of DNA called chromosomes. These genes and other molecular codes in our DNA determine how an organism grows, from a single cell into something far more complex.
Since the Human Genome Project was completed in 2003, we now have a map of the entire human genome. Since then several other genomes have also been sequenced, from chimpanzees to mice, and even bacteria. We now know that humans are all 99.9% alike in their DNA. That is, the many differences between all humans comprises only 0.1% of our genome.
In contrast, humans share about 95% of our DNA with chimpanzees (or more, depending on how it's compared). The more distantly related the animal, the less DNA we share in common.
The similarity between genomes on its own is not sufficient to prove common descent. Creationists like to appeal to a common designer to explain this detail, arguing that God would have reused components in creating each species.
But there is some evidence in the genome that makes common design extremely unlikely.
There are mistakes in our genome, caused by ERVs (Endogenous Retro Viruses) and transposons (similar to viruses, but they only replicate in the genome of the host). These result in viral DNA being transcribed into the genome essentially in a random location. If they happen to occur in germ cells, they are passed on to offspring, including subsequent generations.
As explained above, finding the same transposon in the same chromosomal location in two different organisms is strong direct evidence of common ancestry, since they insert fairly randomly and generally cannot be transmitted except by inheritance. In addition, once a common ancestor has been postulated that contains a certain transposition, all the descendants of this common ancestor should also contain the same transposition. A possible exception is if this transposition were removed due to a rare deletion event; however, deletions are never clean and usually part of the transposon sequence remains. Using the same principles behind DNA fingerprinting, biologists have used transposons, pseudogenes, and endogenous retroviruses to demonstrate that many species are genetically related, such as humans and other primates.
Most importantly, in the human α-globin cluster there are seven Alu elements, and each one is shared with chimpanzees in the exact same seven locations (Sawada et al. 1985).
29+ Evidences for Macro Evolution
This last part cannot be overstated. Scientists understand the mechanism for how these retro-viral insertions end up in the genome. They can identify several of them in the same locations in both chimpanzees and humans (and likewise there are such similarities between other closely-related species). The same methods can and have been used in paternity cases and in criminal forensics. The natural explanation is that we share these same viral DNA sequences in the same locations because both chimpanzees and humans descended from an earlier ancestor that was infected by the virus. Scientists can even use these genetic markers to trace different lineages, including human ones.
All a creationist can do in the face of this evidence for common descent is claim that "God just happened to do it that way". And indeed the same could be said for any evidence, which makes creationism unfalsifiable, and therefore unscientific.
One last, but significant piece of genetic evidence comes from what we find when we use the genetic similarities to map relationships between species. When all of the sequenced genomes are fed into a computer and analysed, the resulting family tree corresponds almost exactly with what had already been determined through earlier methods such as morphology. Charles Darwin, who published the theory of evolution, had no knowledge of DNA, and yet more than 100 years after his work was published, evolution has been confirmed yet again in spectacular detail.
It is difficult to imagine how all of the above could be true, if we were all specially created by an intelligent designer. At the very least we would need to accept the idea that the designer just happened to create all species in such a way as to make them appear to have evolved through common ancestry.
The far simpler, and more natural explanation is that we really did evolve through common ancestry, whether there was any designer behind it or not.
I've barely scratched the surface of the available evidence for evolution. However even just what I've outlined above is enough in my view to demonstrate that evolution has occurred. We could talk about the proposed mechanisms by which it has taken place, but that's a separate discussion. Scientists have a good understanding of the basic mechanisms behind evolution, including how and why it works, but there is still a lot we don't know. But the fact that it has occurred is undeniable given the copious available evidence.
Some creationists try to point to gaps in our understanding as evidence that evolution isn't true, but this reasoning is flawed. You may not understand how an aircraft works, but seeing one flying through the air is enough to conclude that it does. Likewise even if we don't understand every minute detail about how evolution works in every case, we can see the evidence of change over time (or 'descent with modification') and conclude that it really did happen.
If you do want to see a more comprehensive list of evidences for evolution, I highly recommend reading this article: 29+ Evidences for Macro Evolution.
Has speciation been observed in the lab? Yes, it has.
Has speciation been observed in nature? Yes, it has.
Are there any transitional fossils? Yes, there are. And here's another list.
You can also find many answers to creationist claims here: Index to Creationist Claims.
In my experience (and indeed this was true for me also) the two main reasons why people deny evolution are (1) a general lack of education on the subject, and/or (2) a prior commitment to (religious) beliefs that were inherited during childhood.
As I mentioned at the beginning of this article, if your view of evolution is such that no sane, rational person could possibly believe it, then I would strongly suggest to you that perhaps no sane, rational person does believe what you think evolution is. Perhaps they believe something else instead. It's time you read some more about what scientists really do believe, and what the evidence shows. You're not going to get that from second-hand sources.
Remember that the truth does not always conform to your existing beliefs. Any good scientist or truth-seeker knows that they should follow the evidence wherever it leads, regardless of whether the conclusion is comfortable. If you've previously accused others of simply believing what they want to believe rather than honestly following the evidence, then it's important that you don't make that same mistake yourself. Lying to others is dishonest. Lying to yourself is stupid.
Good luck in your search for truth.