Pumping up my GRE score, or getting some publishing under my belt?
I suppose you mean the GRE general test, as opposed to the subject tests. The general test is easy enough to study part time, and there are openings to take the test almost everyday. Plus, you get to send only the best score now, via "Score Select". So, there shouldn't be a problem doing both at the same time.

IMHO research experience greatly trumps GRE scores. Great GRE scores won't help you get admitted and bad scores won't necessarily cause your application to fail. Good research experience, from my experience applying to grad school twice and speaking to professors, is solid gold.

Bear in mind that during the PhD, you will have to juggle coursework, teaching and keeping up with research in order to find an adviser during the first 1-2 years. In comparison, studying for the GRE while doing research is almost trivial. One trick is to not spend too much time actually studying for the GRE. Instead, get a book like Princeton Review, learn the tricks and game the system.

Tl;dr - Do both at the same time!