What parts of Magic Missile are affected by a +1 caster level?
I have a level 1 character with +1 caster level to magic missile spells. Does that mean he casts at level 2 for all purposes? I expect that would include the spell's maxium range, ability to overcome spell resistance, and number of missiles, right?
Does that mean at level 2 he'll treat it as though he had 3 levels in his caster class and gain his second missile?
Yes, you are correct for all your questions.
Just remember that magic missile can't be affected by the Intensify metamagic feat to get more than 5 missiles.