For parts that you can remove and submerge in a small container, I think Evapo-Rust is the best you can do. Given enough time, it is nothing short of witchcraft. The nice thing about this product, apart from how well it works, is that it's safe to use (I stick my bare hands in it with no ill effects) and environmentally friendly. It isn't cheap relative to the budget for a bike like you're working on, but it is reusable. I think I read that a gallon of it will treat 500 pounds of rusty metal (though I personally would have spilled most of it long before that).

Here are before and after pictures of a rear derailleur I soaked overnight in Evapo-Rust:





The dark spots you see in the "after" picture aren't rust. They are places where the chrome plating had flaked away (earlier, not because of the rust removal) and the bare steel was exposed. On non-shiny surfaces, this kind of dark spotting is common. Chrome surfaces with heavy rust are likely to have pitting. Regardless of what rust removal method you use, these aren't going to regain the smooth mirror-like surface they once had. Instead, you'll get a dull, dimpled result. It's better than rust.

As for the frame, I'm still experimenting with various methods. I'm certain that submerging in Evapo-Rust would work for this too, but it takes a lot of liquid to submerge a bike frame. One thing I tried recently was wrapping the frame in paper towels soaked in Evapo-Rust (heavy blue shop towels work much better than kitchen variety paper towels) and then wrapping those in cling wrap. I peeled the wrapping off periodically and scrubbed lightly with a copper detail brush then re-wrapped. Obviously this only treats surface rust, not rust inside the frame, but it worked pretty well. Here are before, during and after pics from about four days of that treatment.







I lost a lot of paint, but I had a lot of rust.

I'm working on another frame right now that wasn't nearly that bad -- just a lot of small spots and patches of rust.



I'm trying naval jelly (available at any auto parts store) with this one. The directions say brush it on, wait 10 minutes and wash it off then repeat until the rust is gone. One thing I've noticed is that this is only taking a layer at a time. I was able to improve the results significantly by scrubbing the spots with a Mr. Clean Magic Eraser before using the naval jelly. The Magic Eraser gets rid of a lot of the piled up and superficial rust. It's my understanding that it isn't safe to leave the naval jelly on paint, but for the 10 minutes recommended it doesn't seem to cause any problems.

Finally, if you do much Googling on this topic you'll come across the recommendation to scrub the bike with aluminum foil. A little more searching will find people saying this doesn't actually remove the rust and is a scammers trick. I tested this out recently, and my conclusion is that for superficial rust on chrome surfaces aluminum foil works great, but for deep rust it's not helpful. Here is the fork I was testing with. The leg on the left was scrubbed with aluminum foil. The leg on the right was soaked in Evapo-Rust.



My "before" picture for this is at a bad angle, but you can see that they started out in similar condition.