When composting materials that are low nitrogen (wood shreds, twigs, brown leaves, etc.) I like to add in materials with higher nitrogen to speed decomposition (manure, "green manure" crops, grass clippings, etc.). You can add fertilizer if you don't have much high-nitrogen green materials available, a good way to use up fertilizer that has gotten too damp to spread well.
[The paragraph above was edited slightly, at the request of the Department of Redundancy Department
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All of my compostables go into a compost bin made of old plastic landscape lattice (photo). I have a big bin (in photo, 40 inch diameter x 48 inches tall), and a small one that is a third of that size. The small bin is my "first stage" bin, when that is filled I flip everything into the big bin to aerate and mix the compost, and decompose longer. The bins are open at top, sides and bottom, directly contact the soil so earthworms and insects can get in to help decomposition. I would recommend such a bin (skip the lid).
In any composter, wood can take a while to break down. If you are not screening the compost to separate coarse materials, you may want to use that woody compost where the wood bits are less important (under your hedge or shrubs maybe OK, but using for carrots right away may not be as good).
