First, Don't bother with buying a plastic bin or tumbler unless your compost pile will be in view and you need to keep it fairly tidy looking. Unfortunately, your pet cage seems a bit small, but you could start with it!
We have used both a Bio-Stack (high-rise composter) and a wire cage composter with equally good results. If you decide to go with a wire cage you'll need a small roll of galvanized welded wire fencing (available at lowe's, home depot, etc...). You'll want a roll that is approx 36"-48" high. Small rolls usually come in lengths of 50', but you'll only need a 6'-7' long section. Make sure to cut the fencing so that you have the ends of the horizontal wires to bend around the opposite edge of the fencing. Like this...
You will end up with an inexpensive and durable cylindrical cage for your compost. It's not pretty, but it works well. This one is 5 years old and has been bent, mangled, and pushed back into shape several times...
Set it away from any building or foundation directly on the ground. Add layers of grass clippings, leaves, weeds, and kitchen scraps. Keep it moist and turn it every 7 -10 days. All of the scraps you mentioned are great, but keep in mind that unless you are consuming strictly organics that some things, like, pineapple, banana and citrus peels/skins are sprayed heavily with all kinds of nasty chemicals and take longer to breakdown.
The photo below is our messy composting corner. We dug a good sized pit at the end of last years gardening season and threw all of the spent annual garden plants, clumps of weeds, grass clippings, partialy composted material from the cage and high-rise composters, and fall leaves into the pit and covered it with dirt. We dug it up this spring and had about 2 cubic feet of nice fluffy black gold (compost) to add back into this years garden soil. The pile of grass (pictured at the base of the 2 cedar posts leaning against the fence) is the result of tilling a new area for a wild flower garden elsewhere in our yard and it will end up as the bottom layer when we start a new pit in the fall.
Good Luck and happy composting!