
Schrödinger's Dragon |

Your caster levels are kept separate from each other, so if you were a Sorc 5/Bard 3, you would have a caster level of 5 for your sorcerer spells but a caster level of 3 for your bard spells. This same logic applies with feats, so only your highest caster level is really relevant for qualifying for feats (or determining their effect, in the case of something like Arcane Strike).
There are a few ways to raise your effective caster level though, most notably an Orange Prism Ioun Stone or Magical Knack trait.