While it can be expensive to learn first aid at the higher levels and skill-up by purchasing cloth from the auction house, first aid is still a very vital profession for all classes to have.
At level 42, my rogue has first aid up to craft mageweave bandages. They can restore 800 life over 8 seconds. That's 100 life every second! The food I can use is level 35 and restores 1392 life over 30 seconds. That's 46.4 life every second.
Using a bandaid instead of eating food will more than HALVE the time you spend restoring life between battles.
Bandages also have the bonus of being able to be used in combat, and they don't consume mana. This is especially nice on a healing class such as a shaman, because then you can use your precious mana to deal more damage instead of running out trying to keep your life bar at full.
As if that's not enough, using a bandaid can be handy when grouped in an instance during an emergency situation. The healer will probably be too busy keeping the tank alive to spare any thought for the DPS. Commonly, DPS players will be expected to step back and bandage themselves before jumping back into the fray.
The only reason you might not want to bandage in a group is if you're the tank. A stingy healer could interpret the tank bandaging himself as a sign that he doesn't trust the healer to keep him up. I would like to send a message to all those healers: A tank may bandage even if you're a GREAT healer! Bandaging is a reflexive skill to an experience player. Do not discourage your tank from doing so, it may save your life someday!