I don't necessarily agree with the one-week courses mainly aimed at women who want to do concealed-carry, but in order to get up to speed, it doesn't take too much longer than that - as long as you are willing to keep up with regular practice, and you are willing to take some time at home to learn how the gun operates. (i.e. take it apart, clean it, put it back together, etc.)
You're probably going to want to learn on a normal-sized gun at first, prior to getting something that's concealable. It's just easier to learn that way. When you're ready to get something concealable (preferably NOT to be kept in a pocket) you can't do much better than this: http://www.kel-tec.com/p32.html
. It's one of the most common concealed carry guns. Just make sure you're following state law wherever you are, as every state is different.