I've had a real close look at some of the possibilities put forward (I lack the imagination to come up with my own) and none of them are viable.
Worm Holes - these exist in the foam of space time as the smallest bubble possible (note I did not say imaginable because it is far smaller than that). To make one the size of a basketball will take the energy of the sun. How do we enter such a thing and live ? It would be like a black hole. How do we control where it goes ? How do we make it longitudinal instead of spherical ? The basic assumption here is that space is curved and it isnt. It is flatter than flat.
Space Warp - this involves squashing space in front of the ship and spreading it out behind it, so space is covered in a shorter distance. This is reliant on understanding gravity and using it as easily as we now use electro-magnetism. I dont know where to begin to tell you the problems with this.
If all problems are solvable, then we still have time against us. If we can travel huge distances through space we will also be messing with time and this will involve all the paradoxes from Einstein's thought experiments. Time and distance are two ways of measuring the same construct...seperation. Nature wants us to keep our troubled little existance to ourselves.
If we just admitted we are stuck here, we might just take better care of the planet we have. But it will be very difficult to live on in a billion years time, if something doesnt happen before then, and it is mathematically certain we will have at least one major asteroid impact before then. So we should have a long term plan to leave. Just dont imagine we will be going very fast, because we wont be.