I'm not the most experienced with troubleshooting smartphones, but random restarts sounds like (to me at least) a problem with the power supply (i.e. the battery).
As a diagnostic step, try taking out the battery and checking the contacts (on the battery and in the phone) for any signs of dirt or foreign objects (e.g. hair) that might be obstructing it.
Also, if you do choose to send it off, try two things before you do. First, take it to a trusted phone shop and ask for their opinions/advice. They're more likely than random forum users to have encountered this problem before, and may know how to fix it. Second, try calling HTC support and seeing what they have to say. It's probably a common problem with a simple fix, or maybe just a very basic hardware fault that takes minutes to fix. They'll probably know best on what course of action is best.
EDIT: Oh, and remember to take out the SIM/SD cards. You won't need to send them in with the phone, since they'll use their own tools.