I put all I can into the compost, I'd rather see stuff be "recyled" than thrown in the trash. All kitcken scraps (exept meat and dairy) go in, as well as paper towels, dryer lint, and floor sweepings. I've even composted cotton t-shirts, socks and towels. Cardboard is also good. From the yard I find all the organic material I can: grass clippings, leaves, garden waste, old mulch, straw, annual plants in the fall. I agree that dog and cat poop should be left out, but horse, cow, rabbit, and chicken manure all quite safe to add.
I just pulled up all the old plants from the garden today and threw those in. These alone filled up two of the three bins. Along with these I layered in straw and leaves to avoid a stinky mess later on. Once these settle a bit, I'll combine them in the third bin with shreaded leaves mixed in and I'll water it well every few layers. When the pile is complete,I have a 5' length of rebar with a point ground on one end that I use to create air holes. This gets the pile HOT. Sometimes too hot.
My "secrets" are to keep the pile moist, aerated and turned when it cools off. I can generate a lot of black gold in one summer.