If you want to try your hand at making cheese at home, the easiest one to start with is Ricotta. Ricotta is traditionally made with the whey left over from making mozzarella. In fact, the word ricotta in Italian means "recooked" and refers to the fact that the whey is cooked and curdled again after the first batch of cheese is made. However, it is also possible to make ricotta directly from whole milk, and even store-bought milk works well (whereas with most other cheese, store-bought milk will not curdle properly).
Everything you need to know about eating and cooking with curds
Most milk-based methods involve heating the milk up to a temperature just below the boil (about 180°F) and adding a coagulant, typically lemon juice or vinegar. Once the milk separates into curds and whey, it is strained through a cheesecloth and a bit of milk or cream can be added to get the proper consistency.
Chef Michael Chiarello's Napa Style website details a method I've never seen before, using buttermilk as the coagulating agent. I haven't tried this variation but I suspect that the buttermilk adds a nice sourness to the final cheese without drastically affecting the overall flavor. Using lemon juice or vinegar can contribute in obvious ways to the final flavor of the cheese. This is not necessarily a bad thing, but if you are using the ricotta in a dessert, you probably don't want the vinegar taste to feature so prominently. And a lemon flavor might not be what you're going for. In that case buttermilk would likely be a better choice, melding more gracefully with the milky curds and adding just the right tang for a sweet final course.