I'm studying category theory and, given that I don't have a background in topology, I'm struggling to think of some finite categories that interesting.

The main one I know of is finite preorders -- I find this category interesting because it has products, sums, etc. I'd love a couple of other categories like that that I could use to better understand (by graphing out the objects and morphisms) things like equalizers, pullbacks, etc.

I could of course do this by construction, but I'd much prefer some finite categories that include these ideas in a useful way.

Thank you!

nota finite category, so I'm not sure what you're really after. The Monster group is an interesting finite category with one object and quite a few morphisms. The poset of truth values $\{0 \leq 1\}$ has a lot of nice properties (e.g., being cartesian closed) that are worth contemplating. Something that you might wish to contemplate is why the category of finite categories doesnothave coequalizers! $\endgroup$