For the past several months we kept our four bikes in a rather damp shed. Not surprisingly, exposed steel parts on all of the bikes have started to develop small rust spots, including chains, cogs, and chipped braze-ons.

What are the best ways you know to remove rust and repair damage?

I was surprised that the chain on my best bike was affected. I thoroughly cleaned and oiled it just before I put the bike in storage.

For localized rust spots, I would remove them with a dremel tool.

Then paint with primer, and a top coat if you wish.

My last steel bike was covered with small primer spots. Made it look more "urban" and unappealing to thieves.

So where is it now? It got stolen...

I served as a Marine on Navy ship and I saw sailors work miracles with Navy Jelly. Google that for procedures and sources. If your chain still rusted after oiling, there was moisture in places where the oil didn't cover. A good way to get rid of moisture is using WD40 because it displaces water. Let it dry, then lube. Be sure its dry or the WD40 will break down your lube and make it worthless.

Rust on chains, cogs and brazeons is not a problem. Just lube the chain with wet stuff and the rust will dissapear. You can spray the cogs with WD40 but be really careful, WD40 can get inside the bearings and dissolve grease. As for brazons, just wipe with an oily rag.
I use car wax on the frame to prevent rust.
Check inside the frame for water collecting in the BB shell.

Do not do that! Most dremel bits other than the wire brush are liable to punch a hole through the frame, and the wire brushes are much too small to mess with. Perhaps if you are really careful and good with dremels you could get away with it, but without any offense to to you cantor, the very fact you had to ask leads me to believe you shouldn't use a dremel. What I would do is get a fine grit diamond file and use that. Then as eubi said, prime and paint.

As far as the chain and stuff, I'd pull them off to protect the frame, then use strong cleaning chemicals from the hardware store. One word of advise, however much time the the bottle for the cleaning agents say to wait before you spray the parts off with water, follow it exactly. Acids can disentegrate metal fairly easily, I wrecked 1 or 2 window AC units in my HVAC days using acid compounds to try and restore them.

I'm surprised no one has mentioned this yet. But after you take care of the rust spots you should find a better place to store them to prevent it going forward.

Good point. Thanks for the clarification.

I have a very light touch with a dremel and use fine grit sanding wheels on a rubber spud.

Then again, you would have to do some serious sanding to punch through a frame!