When your compost isn't decomposing fast enough for your liking, what you really need is a tumbling composter. Specialty shops carry them, but they're generally expensive. It's relatively easy to build one for a fraction of the cost. The simplest ones can cost you nothing more than a bit of your time, depending on whether you've already got the materials you need in your house. At most, you may have to make a short trip to the hardware store or phone around to see if anyone's got a spare barrel.
Narrow down your potential compost bin options. If you've got a large, disused plastic garbage can sitting in your back yard or garage, this would be a perfect use for it.
Aerate your new compost bin. Decomposition can only happen if air and moisture both have access to your compost heap. However, you don't want the holes to be too big, because otherwise your compost can escape all over your yard before it's fully decomposed. Also, undesirable pests can take up residence there. Use a drill, hammer and nails, or a nail gun to poke many small holes all over the bottom and lower half of the bin.
Fill your new compost bin with kitchen scraps and other household waste you wish to compost. Sprinkle it with water, make sure the lid is secure, and give it a good roll around your yard.