Hi, Deborah,
1. What is different about jasmine rice? It tastes like any brown rice to me.
It's a variety of aromatic rice that is fragrant, smelling a little to me like flowers, and that's why it's called that.
2. Likewise, does basmati rice taste any different?
Basmati is another aromatic rice. To me, it smells and tastes a little nutty.
3. Is short or long grain more tasty or something? Is there any difference other than size?
Short grain rice is a little stickier/clumpier -- with long grain rice, the grains stay a little more separate. The flavor is similar, but the texture is quite different.
4. Is sticky rice a different type of rice, or is it cooked differently to make it sticky?
Two-part answer: any short-grain rice can be cooked in a "sticky" fashion; it just clumps more. But there's also a kind of rice called glutinous rice (not related to gluten) that is extra-starchy, extra-sticky, and used when the stickiness is the quality you're looking for.
As a former chef, I recommend that you experiment with all kinds of rice; you'll come to appreciate the differences. However, if your rice is not from California or Asia, I recommend you cook your rice using the pasta method -- that is, boiled in plenty of water, instead of the normal steaming method. This is to reduce the risk of arsenic.