You can get a good start by putting your lawn clippings from mowing in the composter. They are some of the easiest yard waste to break down. Run the hose over and spray it occasionally, and let that grass get hot--literally. Compost, when it's working, will give off steam on a cool night. Weeds pulled from the garden (include the dirt from the roots--there are lots of micro-organisms that help with the composting) and table scraps of the veggie variety will all compost nicely. Putting in grass regularly helps keep it hot. If you water it a couple of times a week (don't know how big it is, but you need to give it a good soak) and every week or so (this is a guess--I am not so consistent as to turn my compost that regularly) you will allow new material to get into the middle to cook, turn in some oxygen, etc.

You don't need some fancy compost-starter (I've seen stuff for sale that is supposed to jump-start the compost). Just use the grass.

SRS