Ideal composting is basically a nitrification cycle taking place inside your "bin." This can be difficult to achieve, depending on what you're working with, but chances are, no matter what, your stuff is going to break down over time regardless over whether you have put together the optimal blend of carbon and nitrogen or not. Basically, you want a mix of your greens and your browns, preferably with a higher percentage of browns. Your browns include items such as: leaves, newspaper, corn stalks, peat, pine, sawdust. Greens are food waste, garden waste, coffee grinds, weeds (as long as they haven't gone to seed). Do not ever put any type of animal product or animal waste, citrus, chemicals (herbicides, pesticides).
This is the composter I currently have, which makes it easier to turn the compost-
I have also used this one, from Lowes:
I find that the first one heats things up a little more quickly and I prefer to not have to take a pitchfork or rake to it to turn it, but to each his own. These can be super easy to make, as well, if you have access to tools for cutting wood. There are tutorials all over youtube on different ways to make these.
As to collecting the compost itself, I used to use just a bowl on the counter, but as that can get a little gross to look at some times, we have recently introduced a compost crock. This one that I have is from Plow and Hearth.
I think this one is very pretty, but there are a lot of options! Many have filters in the top to keep them from smelling as badly, and also compostable bags to line the inside with, so that they are easy to keep clean.
Honestly, I haven't used mine much in the garden, but it is amazing every week when I go out with my load to dump on top, that there is amazingly little in there! (Maybe I have a neighbor with a very fertile yard?) Good luck with yours! Get composting!