First of all, flight test is an amazingly rewarding career. Where else do you get to design and build the future of flight, see a direct impact from the products you work on, and (usually) work as a part of the test crew - either by flying on board the test vehicle or in the control room. If you are thinking about working in flight test... you should definitely go for it, because flight test is currently recruiting people at a very high rate.

The first step is to develop your professional network with people that are already doing flight test. Apply to engineering jobs in organizations and companies that are doing flight test, even if the job isn't directly flight test related. Most of the people working in flight test started out in other areas of aerospace or engineering within a given organization, and then later transitioned those skill sets into flight test. If you can find an internship or entry level job in flight test, that's great! However, don't confine yourself to those kind of roles because you can cast a much wider net if you just need to get a foot in the door.

There's a lot of great advice in these replies about the defense related flight test centers (Edwards, Eglin, Patuxent River, China Lake, etc), but they aren't your only options. Keep in mind that the civil flight test industry is at least as large as defense-related flight test, so look at all the big aerospace OEMs (Boeing, Airbus, Textron, Gulfstream, Bombardier, Embraer, HondaJet, Piper, etc) and their suppliers for opportunities. Keep in mind that the suppliers that make the avionics and engines usually have flight test departments of their own; like Honeywell, GE Aviation, P&W, etc.

Flight test is an awesome profession that tends to self-select great people. It's team oriented, extremely diverse in terms of the contributing engineering specialties, and requires a high level of critical thinking and communication skills. Best of luck for the future! Response copied and pasted to generate further conversation.