Can't we use genomes to compare species and then postulate how evolution has progressed?
We can and we do, there are hundreds of studies looking at the comparative genetics of populations of one species up to large genetic comparisons across dozens of species or even more.
The process would probably be tedious, but I'm sure it'd be accurate.
As the methods used to collect and analyses large volumes of genetic data become simpler and more reliable the tediousness is reduced. The accuracy is to a large extent dependent on the method of analysis and the degree to which the assumptions which are used in such analyses reflect the reality.
Like you said, however, I would rather see evolution in the fossil record so I can actually trace a particular trait or characteristic as it evolves over the years; this would aid my imagination.
As the ability to recover DNA from more ancient tissues improves this has become more feasible for genetics but there is never going to be a comparable level of preservation to that seen in gross morhphology, or even cellular morphology in some cases.
The best thing to do is to look at both the genetic and morphological data (such as fossils) in parallel and the degree to which the trees of relationships between different organisms agree from these two independently derived sets of data seems to many a compelling point in favour of common descent.
While the derivation of these data is independent they are fundamentally connected as the morphology of an organism is largely dictated by its genome. The relationship between genetics and form is not a simple one however and directly connects to some of your questions about the nature of mutations.
I've learned that the factors that cause mutations are called mutagens, but I'm still slightly confused as to how often these mutagens happen. There must be a tremendous amount of mutagens and mutations to account for the diversity of life.
There are, mutagens are all around us constantly. Every time you see a new report in the media about how something increases your chances of developing cancer they are talking about that substance's potential as a mutagen. Sunlight is mutagenic, atmospheric background radiation is mutagenic, oxygen is mutagenic, red meat is mutagenic, salt is mutagenic it would be well nigh impossible to remove all the potential mutagens from one's environment and still have an environment in which life could exist.
2) These mutations account for the vast variety of characteristics that see in the natural world, right? From a whale's blubber to a bird's feathers, mutation must have been the cause of it. I still don't really understand how that can happen, exactly, but I feel that I'm getting closer.
That is correct and Crashfrog pointed you to the relevant discipline, evolutionary and developmental biology, which concerns the evolution of the way in which an organism develops. With modern genetic tools the degree to which the effect of mutations impacts the morphology of an organism can be studied to a degree previously unthinkable. The effects of changes in a single base pair of DNA can be studied in terms of changes in the morphology of the organism and changes in other genes governing its development.
Similarly the genes, and indeed whole genomes, for differing species can now be compared to allow the identification of the genetic differences between those species which can highlight possible genes or genetic elements responsible for the morphological differences between those species.
TTFN,
WK