My dad has an old Model 1906 22 long rifle/short. It has some problems feeding. Sometimes the cartridge does not want to go into the chamber and half the time it takes two or three pumps to get a bullet out of the magazine (the long tube thing). I would like to fix it up without anything getting to complicated or expensive. Any tips on where I could start?

Go to a gunsmith and pay him to fix it.
By doing it yourself, you run the risk of breaking things and parts are hard to come by for these old guns.

-Spencer

Your Dad's rifle is a takedown. If you separate the receiver and clead out the crud
you may fix it. Be sure that the "flipper" in the elevator is free to move up and down. Do not try to take anything else apart. If you still have trouble, take it to a gunsmith.
These are classic guns, and deserve care.
Good Luck!