Agree with Maximus. Do the TSA fingerprinting immediately upon arrival in the US. It's much cheaper that way (ballpark figure 40 vs. 400 euro) and if you make sure it's done electronically, it may only set you back 24 hours or so. Which can be well spent in doing the ground exams for instance.
Doing them in Europe is a very time-consuming hassle and only really useful if you are going for a type rating or something similar, where you can only start training once the TSA has finished the whole vetting process - which can take up to 30 days. But a PPL student can start training as soon as the TSA acknowledges receipt of the fingerprints. Which is normally a matter of hours if done electronically, and a day or two if done on old-fashioned paper fingerprinting cards.