"If we were able to build a spaceship that could travel at the speed of light, what would be the physical consequences? As an example, I think Alpha Centauri is about 4 light-years away, so I basically need 4 years on that spaceship to reach it. But, will my body age at the same rate when I am on the ship or will in remain the same?"
Firstly, the physical consequence of traveling at the speed of light is that your mass becomes infinite and you slow down. According to relativity, the faster you move, the more mass you have. The same works on Earth when you're driving down the freeway. You weigh a tiny bit more driving around than you do when you're completely at rest.
So, traveling at the speed of light in the conventional way is impossible. Fortunately, nature offers us a few ways of cheating. These methods include wormholes and traveling at warp speed, among others.
To Further Explain:
As far as I know, time dilation (which is what your question is centered around) doesn't apply in any of these cases (or, at least at a much less-noticeable for things like warp where you are still in 'normal' and 'non-relativistic' space - again, as far as I'm aware). So, I'm going to skip these and talk about traveling really fast using 'conventional' means.
Time dilation is "an actual difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from gravitational masses". This is why astronauts returning from the International Space Station have 'traveled' 0.007 seconds into the future after spending 6 months in orbit. Of course, they haven't really traveled forward in time; they just aged slower relative to everyone on Earth. So, how about I use Alpha Centauri as an example. Lets assume you are able to instantly accelerate to 99.9% the speed of light, as well as instantly decelerate, without killing yourself or destroying your ship (which would probably result in your death).
Alpha Centauri is technically 4.5 light-years away, but Wolfram Alpha won't let me do that calculation, so I've set the value for 4. Traveling at 99.9c for 4 years (Earth time) means you'd experience a trip of about 2 weeks (14 days). Yes, everyone on Earth will have aged four years and you will have aged only 2 weeks. To imagine this dilation effect for this particular trip, you can think of it one of two ways; either time on the ship is moving 110 times slower than ‘normal’ or you’re traveling a distance 110 times shorter – as far as spacetime is concerned, it’s the same thing.
Of course, communication from that distance means any signal you send (assuming it was powerful enough to be heard and understood) would take 4 Earth-time years to get back to Earth, so it's probable that you'd show up in the system, do science for a year or so (likely sending some type of signal to Earth to let Mission Control know you’re still alive – even if you can’t transmit usable data), then return to Earth. You will have been gone for a total of 9 Earth-years and about 1.24 years your time (this is excluding the time dilation effects of not being in an Earth-like gravity well for the duration of your stay in the Centauri system).