If you feel like spending money, I would wager you might prefer a new frame since you ride a fixed gear and might notice improved stiffness. If you're set on spending money to go faster, aerobars are probably your best bet.

edit: was typing during your last post.

Investing in the frame or wheelset first would probably depend on what was on your existing hypothetical geared bike.
Keep in mind that bikes & parts are usually cheaper if you buy everything at once as a full bike.