Skip composition or not?

If you have either an ice/lightning weapon, coldbeam/thunderspear both beat comp on neutrals respectively, so there's really no point in getting comp if you have one of those weapons. If you don't, there's really no point not to max comp since it'll be your best single-target attack until 4th job, unless you plan to max slow/mp eater from 2nd job over it.

First of all, try to get your booster to 11 and amp to 30 ASAP.

As for training spots, you could try gobies or showa's bains.