There are all sorts of different rust removal techniques. Which to use depends on what part(s) we're talking about.

Can it be submerged? If it's a small part like bolts or nuts, dunk it in Evapo-Rust overnight. If it's a big part like a frame, dunk it in an oxalic acid ("wood bleach") solution overnight. Pull it out in the morning, rinse, and dry. Easy and effective, but you may not want to go this route unless you can completely disassemble. Parts with internal (lubricated) mechanisms, like freewheels for example, aren't good candidates for submersion.

If it can't be submerged and you're just looking to treat surface rust, you can go at it with vinegar like John said. Or Evapo-Rust, or naval jelly, or a few other different items.

Once you've got the frame rust-free, you can optionally treat the inside with a rust preventative. J.P. Weigle's Frame Saver, Boeshield T-9, and AmsOIL Heavy Duty Metal Protector are all popular choices.