the fact that you had the foresight to ask this question (means your thinking ahead), and even come up with the right assumption, makes me think you'll do absolutely fine on this trip.
If you haven't been to a class C airport before, it won't be all that different from a class C or E controlled field. Just stay on top of the controllers instructions as you depart because it is not uncommon for them to assign you headings and altitudes to follow until your clear of the airspace