Brush trimmings aren't greens, they are carboniferous browns (but your grass clippings are greens, so that's ok). You do need to break them down considerably to put in a compost pile, unless you want to keep a separate brush pile somewhere. The brush pile will collect leaves and be good critter habitat and will very slowly break down over a period of years. To put branches and roots and other woody stuff in your compost pile, it needs to be well broken down. I have a little chipper-shredder, I got cheap on ebay. It's not high powered enough to run thick branches through, but it's great on all the woody stems and tough vines (tomato plants at the end of the season), heavy roots, etc. I let them pile up for a little while and then get out the shredder and chip them up to go in compost pile
The classic book on the topic is Let it Rot! Gardener's guide to composting
by Stu Campbell. But pretty much any good gardening book will have a section on composting.
Speed - thanks for the compliment. It's nice to have a place where I can pass on some of what I've learned from 17 years or so of doing this stuff, even though I don't have any certificates etc in gardening (maybe someday I'll take the master gardener classes).. Don't dig your hole very deep, just a little to loosen up the top layer if you want. If you make a pit, then you are doing pit composting, which is anaerobic. You want air to get through. And your compost wants to stay slightly damp, but not wet. If you dig down to where the ground is really hard, then rainwater will pool down there.
Shredded paper + kitchen scraps (including used paper towels, coffee filters with the grounds, egg shells, etc) + grass clippings is a beginning. Pulled weeds are good for more greens and coarser texture to keep things from compacting, keep air channels through the pile (surely you have weeds!?
). Fall leaves. If you don't have a lot of weeds and leaves, beg some from your neighbors -- they will think you are nuts, but gardeners are used to that.
Watch curbside for bags of yard waste put out and steal them (works well in the fall when people are getting rid of leaves, you don't necessarily want people's branch trimmings). Well composted manure is a big plus if you can get some and those handfuls of dirt I mentioned earlier. Variety is the spice of life and compost!