First you get a tub like this:
Then you drill lots of holes in it. The holes are about the size of my pinkie, maybe a little smaller. Make them in all the sides, the top, and the bottom too.This is the stuff I put out this morning. There are all kinds of lists people have posted about what is and what is not compostable. Mostly, it's all organic matter that isn't dairy or meat.
To have successful compost, you need to have about equal parts brown matter and green matter. You can think of it like this: the stuff from your kitchen is green matter, the stuff from trees is brown matter. Some examples of brown matter are: sawdust, shredded bills (this is very satisfying), torn up newspaper, dead leaves, etc.
I keep the small tub of sawdust next to the compost and pour in a couple handfuls every time I add stuff from the kitchen.
Then you shake it all up. Air is very important to the compost, as well as water. You may have heard that Tucson is a desert, so I wet down the compost about once a week. It should be as wet as a wrung-out sponge. If you live in a place that is more humid, you may not need to wet it down.
Your compost will have arthropods in it, this is a good thing. If you can't handle that, composting is not for you. It doesn't smell if you're doing it right. I did have a period where mine was attracting a lot of flies, but I don't think I was putting in enough brown matter, and it was right after I made jam, so the compost was like 80% strawberry tops and hulls. I haven't had much of a problem since then.
Also, notice at the bottom of the last picture, I put a sun chips bag in there almost 2 months ago, and it's still fully intact, crinkling as loudly as ever.