Well, according to a lot of Einstein's work it's impossible to actually travel at the speed of light. However, you could get arbitrarily close to it, so it's still a valid question.
Presumably you could only travel at such high speeds in areas consisting of mostly vacuum; you are correct in that hitting *anything* of substantial mass ('substantial' being denser than gas or dust, essentially) at that speed will destroy pretty much anything we're capable of building. Fortunately, most of 'space' is, well, empty space, or filled with only low-density gas (which doesn't present much of a problem). Several science fiction writers (and many astrophysicists) have actually thought about this; some have suggested coating the front of your ship with ablative shielding (such as water ice, which is cheap and easy to manufacture), and replacing it after each trip. 'Star Trek' does away with this by having fantastic energy shielding, something that is unfortunately not an option (at least today) in the Real World (tm).