If I were to begin play as a 1st level wizard, and lets say I pick up magic missile as one of my starting spells. If I happen to survive till level 2 and become a magus for instance, as a magus I get magic missile as well. So then would I have the potential for 1 level 1 magic missile spells + bonus spells for each class?
If so, what is the effective caster level on Magic missile? Is it 2 or 1?
Yes, if you wish to do so, you could prepare magic missile in all your spell slots for both the wizard class and the magus class.
The caster level is for each class individually, so the caster level would be 1 for both your wizard MMs and your magus MMs. If your third level is a wizard level, then your wizard MMs would be at caster level 2, while your magus MMs would still be at caster level 1.
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HERE are the PRD definitions of "character level", "class level", and "caster level".
Although your character level as a Wizard-1/Magus-1 is 2, your caster level in each is only 1. You Wizard spell effects will go off of your Wizard caster level, and your Magus spell effects will go off of your Magus caster level.
There are some feats out there (and one trait) that can boost your caster level slightly for when you're doing a multi-class caster build.