First, I should note that I already believed in evolution before I became truly convinced; the way our educational system typically works today, we have to believe things based solely on our trust of our teachers long before we can see them properly proven. This, alas, is as true in mathematics as in biology. But since being taught evolution in school, I've had the time to examine just what the arguments are for it, and I am convinced that I was not misled.
When I was reading Darwin's Origin of Species, the arguments that really struck me were by no means certain. They were, however, enough to show that the evolutionary theory was worthy of further investigation. The diversity of living organisms, their apparent arrangement into families, the apparent occurence of mutations of some sort in nature, the successes of breeders, the harshness of the struggle for life--all these combined to make evolution appear a compelling and elegant explanation of how the different species could have been created.
What remained to be examined were the detailed mechanics of how evolution could work. I started my reading with Mendel, who, though his research was in a fairly limited area, did enough to convince me that traits might actually be passed from generation to generation in discreet units, rather than being blended and averaged, or random and unpredictable. As I read later research in genetics, this idea became more and more convincing, and more and more detailed. Eventually, Watson and Crick were even able to establish that they had discovered the physical mechanism behind heredity.
But even before the discovery of DNA, much work had been done explaining just how genetics could explain the workings of evolution on the immediately functional level. I think what convinced me most--the moment I really felt I could accept the theory of evolution based on my own understanding, and not on faith--was a paper by a mathematician, G. H. Hardy, pointing out that, with what we know of heredity, the genetic makeup of a population will stay relatively stable, with no traits become more or less common, unless there is some particular survival value to one trait over another, in which case it will spread throughout the population in a surprisingly small number of generations. This, together with the fact of (and the diversity of) mutation, is enough to establish that, at least after the development of sexual reproduction, the fact that evolution will take place is simply a certainty. Of course, I understood that this was not establish evolution in general with absolute certainty, but it was enough to establish that evolution had indeed occurred, and might be sufficient to explain the diversity of living things.
I myself have not pursued the details of the arguments for evolution much beyond this point. My reading was enough to establish that the theory had a solid grounding, and that, by and large, the scientific community could be trusted in their investigation of the details of that theory. I do occasionally browse more recent papers, just to see how the theory is developing, but overall I am convinced that any changes to our understanding will come in the form of further refining our understanding of evolution, not of overturning it altogether.