Relativity does not permit particles with mass to go the speed of light. However, the math works just fine with massive particles going faster than light. Further however, the effects are quite bizarre, even if well defined. You know how your physical intuition is basically useless when it comes to understanding relativity? It's even more useless when understanding what the math says about tachyons.
Just to be clear, I did say "approachable". I've seen heavier duty explorations of what tachyons are like under real relativity theory, but I don't know of anything else that approachable.
One of the more approachable discussions on the topic is actually http://www.theforce.net/swtc/hyperspace.html . Yes, I am aware of the context, but the author knows what he's talking about. http://www.mssl.ucl.ac.uk/~cjs2/