I don't know much about it. I've heard it's nearly unanimous among physicists that it's impossible to go faster than light. (those speeds are limited to tiny particles)
A somewhat related joke from Futurama that was pointed out to me the other day. Fry wears "Lightspeed Briefs". Fruit of the Loom underwear commonly abbreviates itself as "FTL" an acronym also claimed for faster-than-light-travel. What a bunch of nerds those writers are!
Anything that is not quanta (light particles) expands as it speeds up, the faster it goes the bigger it gets, so a space ship would theoretically expand to the size of the universe at the speed of light. My answer would be that it is not possible at all, unless you subscribe to the varying-speed-of-light theory, or wormholes.
Actually, you are wrong on that. Particles with mass do not expand as they speed up, they gain mass which is very different. For all the extra mass they gain, they do not change size at all.
What you are talking about is when objects made of mass accelerate, they increase their mass. The greater their speed, the greater their mass. The closer an object gets to the speed of light, it's mass approaches an infinite value.
If an object was able to reach the speed of light, its mass would become infinite and therefore more massive than the Universe.
Warp drive, as theorised in shows like Star Trek, do not require an object to move faster than the speed of light. Current understandings of physics do not allow faster than light travel (FTL), but there are other theoretical versions of physics that allow you to get around this. Whether they are applicable to our Universe or not is still open to debate.
Personally, I would prefer to say, "Never say never."
The warp drive discussed doesn't involve the ship actually going faster than light, in effect the bubble of spacetime around the ship is warped, expanding and contracting it, allowing the ship to effectively travel faster than light, while never actually approaching light speed. To somewhat inaccurately borrow from Futurama, the ship doesn't move, the universe moves, taking the ship with it. :D
However, recent studies hint that this particular form of FTL may not be possible after all. Aside from the massive power requirements (the output of a decent-sized star), it has been uncertain as to what quantum effects there may be as spacetime is distorted. A recent study on the quantum effects show that such a warp drive would become unstable as it approaches superluminal speeds.
I'm not well versed on the subject but I've heard that wormholes could sort of provide faster than light travel. It wouldn't truly be faster be faster than light because light through the wormhole would still be faster than any sub light speed vehicle going through and locally the rule would be preserved, even if the ship was moving faster than light outside the wormhole. The problem from what I hear is that they constantly collapse and you'd need to find one and manage to keep it open for a long enough period of time.