Lawn clippings I got. I'm trying to figure out how to compost table scraps. As I said it will take us weeks to come up with enough of that sort of thing and I don't believe you can just keep adding a little at a time.
Spaghnum Moss is a great addition to compost. So is Worm Castings. If you don't mind spending money.
But as for "free compost" I been using table clippings or a while and it's worked our great. Pretty much anything that's not meat, I bury.
I'm not an expert but these are some things I learned by experience and work for me.
First, manually break it down as much as possible, like either blender or a coffee grinder.
Try to dry or dehydrate is as much as possible.
Make sure it's not too condensed in the soil ; mix it up with the other compost evenly.
Bury food deep enough so it doesn't attract flies and such.
Remembering the difference between how long it takes for some things to decompose as opposed to others ;
The longer it takes to decompose, the deeper I bury it, OR the further ahead of time I bury it, before planting.
WORMS. Whenever I find a live worm, he gets relocated into one of my plant soils.
Some examples for me, are like the skins of produce : Sweet Potato skin, Avocado skin, Banana peel, Egg shells ground up into a fine powder.
One way, is take a large cauldron with a matching lid, line it with a plastic Walmart bag, and after each time you eat, throw in the table clippings
into it and slap the lid on until you're ready to compost.
Doing this has so many benefits, besides the obvious ones in the garden ;
You will go through a lot less kitchen trash bags, the kitchen sink and trash can will be more sanitary and smelling good, and dumping less is better for the environment