Hey, I gotta plug Times Up. The little I know about wrenching I learned there. They helped me true my wheels and overhaul my cups and bearings. I learned to change cables, though I haven't had to do it yet. My Brompton doesn't have a derailleur, so that's not an issue. I learned to change the chain and sprocket by myself. In the Lower East Side space, they have classes on Mondays, and open shop on Thursdays. The classes are useful, and the guy who runs Thursday evenings is quite knowledgeable and patient. I ponied up for a membership to help pay back the help I got. I think it helped that I went at the end of winter so I got lots of personal attention; I would guess that they're more crowded now. Warning, their space and their equipment are very, um, pre-gentrification, shall we say (I miss the old Lower East Side)?
You might also want to check out Recycle a Bicycle (also nonprofit) up the street on Ave. C. I don't know what services they offer, but I stopped in to use their pump. My tire kept bulging and the bead wouldn't seat properly, and the guy there helped me fix it, no attitude, no charge.