Here are the things you can compost:
veggie and fruit peelings and parts
coffee grinds
coffee filters
tea bags
tea bag papers
egg shells
leaves
pine needles
grass
chicken poop
cow poop
paper (sheet and news, I shred mine and also use it for chicken bedding)
feathers
garden clippings and debris
weeds that haven't went to seed
I don't worry about speeding up the process. I just toss it all in, turn it occasionally, it rains here a lot so I skip watering it (it must be kept moist), and wait. For this compost corral design, you shovel the finished compost from underneath the bottom landscape timber. I have a video of this corral being made by someone else on a previous garden post.