My bathtub is located on the second floor on an outside wall. There is an area along the outside wall that is recessed, therefore, the area that protrudes out from the house contains the pipes. The area around the pipes has no insulation, so there is no barrier to block the cold between the pipe and pressed wood. My pipes froze in my bathroom last winter and I could not get water out of the bathtub faucet. I am able to access the pipes through the joists in the basement. How can I protect those pipes to keep them from freezing?

A pipe (on the first floor) inside the garage wall,leading to the outside spigot did burst. And that pipe has no insulation between the pipe and pressed wood. I am able to access the pipe since I tore out the drywall. What should I use to protect the pipe and add a barrier between the pipe and the cold?

Any insulation is better than none. Probably the best way to insulate the cavity is to have insulation blown in. Nails from the exterior, pipes, wiring, and anything else that might be there will prevent you from trying to shove any rigid insulation in there, and if the cavity isn't filled, it doesn't do much good anyhow. I would contact an insulation company and have them determine the best way to tackle the problem. It is their area of expertise, and they can do it faster and more effectively than anything you could try. As for the garage wall, as long as there is a frost free faucet and it has the ability to drain, fiberglass batt insulation will work fine. You could use foam pipe insulation for that, but mice would probably eat that given the chance.
Fiberglass helps to insulate and soundproof as well, so it is an extra bonus.