I toss just about everything in mine, all the vegetable scraps that I don't use in my cooking (onion skins, pepper ribs/seeds, potato peels etc), old bread, pasta etc.. I don't have a problem with pests. The grass clippings from mowing the yard far outweighs the stuff that might attract pests, and I always empty my kitchen compost bucket into the center of the pile and bury it.

I have 4 39 gallon trash bags filled with ripped cardboard that I add when/if the pile starts to get slimy. I don't cover my compost pile, either. I'm in no real hurry to turn it around and get it in the garden so I just let it do it's thing naturally. It only really gets turned if I have to add cardboard and to an extent when I empty my compost bucket.

Wont rats and other things get to all those things not covered?

Haven't had any problems yet. Been a couple of months so far since I managed to build up an actual pile. Like I said though, grass clippings makes up the majority of my compost pile, I hide all the 'food' in the center, under a layer of grass about a foot deep.

No meats, no eggs, no dairy, no oils, no salt. You need to have the right ration of greens to browns, and the more you mix it, the faster it will break down. If you do a google search, you will find some good how to's on composting. Rats could be a problem without it being contained, so you may need to fence that area in to prevent them.

A fence won't keep rats out! LOL
You might want to form some sort of bin...maybe with free pallets or some wire in a ring. I've found them at recycling dept of the dump for as little as $5. This will help you bury anything that might attract vermin such as pastas, breads etc.
I don't do meats or cheeses much, but if I do I bury it deep in the pile (that's where the bin comes in handy)
There is a composting forum here with fabulous information and many links to very good info.
Basics-browns about 3/4 is best- leaves sawdust, dryer lint, cardboard, newspaper (avoid the glossy colored paper), shredded paper Greens- about 1/4 -grass clippings, vege peelings and bits, UCGs (used coffee grounds), weeds (i don't do any with seeds ready to sprout. I give these to the goats) Water- keep as wet as a damp sponge. Turn when you want to, or never!
Many people try for a hot pile and turn when it starts to cool, but I've never really tried for a hot pile and still end up with a pretty good stash I put on the beds each spring. I have an old bullet tumbler that I add things to and tumble once a week or so. For me, it seems to break down faster this way. Then I tip it into a regular bin to "finish", turning from time to time.
A few don'ts would be meat, cheese, breads, pasta that may draw vermin. Diseased plants that may carry the disease into the next season.
I lost my main stash of leaves a few years ago, so now I have become the neighborhood leaf whore, coming by the neighbor's yards and vacuuming up their leaves with my free from a yard sale leaf shredder. I store these in bags in the shed and add throughout the winter as needed (I'm in Ca with no major freezes)
Check out the composting board! You'll be hooked! Nancy

We dont have rats here and they may be a different story, but we have plenty of chipmunks, squirrels and mice. I have never noticed them in my compost pile. Its on the side of the house, 2, 4x4x4 plastic fenced squares. I lay down a few layers of cardboard at the bottom each spring to keep the tree roots from growing up into it, then pile on the yard waste and kitchen scraps (fruit/veg/coffee grounds/eggshells) all season. In the late fall add it to the finished beds.

This question is discussed in great detail over on the Composting forum here. That forum - since composting is its focus - has extensive FAQs available on all aspects of composting.

Dave

Here is a link that might be useful: Composting Forum