Warping space isn't such an unrealistic idea. The primary roadblocks are energy and containment. Space is warped all the time. By hydrogen atoms swooping around at near c velocities and by planetary masses like Jupiter. Stars, neutron stars, black holes - all of these things warp space to some extent.
The idea of us compressing space ahead of a ship and expanding space behind it to induce propulsion isn't such a crazy idea either. The same basic principle is what allows planes to fly. Air below a wing is denser than that above (by the shape of the wing pushing through it), and it provides lift. The energy to manipulate that situation comes from the plane's engine.
The problem, of course, is technological. How do we manipulate space around a ship, and how do we do it without harming the ship. Probably via the use of very strong, artificially produced magnetic fields. Of course, we don't know how to produce energy on that scale - especially within something as small as a space ship. That would involve cracking matter/antimatter interactions, or drawing zero point energy [http://en.wikipedia.org/wiki/Zero-point_energy] in large quantities. Which itself involves a greater understanding of quantum science.
And a greater understanding of quantum mechanics could lead to other theoretical methods of transportation which could bypass normal space altogether.
Should we research these technologies and head out to other stars? Absolutely. The technological breakthroughs that would happen in the process of figuring out FTL technology would have dramatic impacts on life on Earth. Tapping into Zero Point energy could mean free, clean energy for the whole world. Matter/Antimatter would be good too, but a bit too similar in danger-factor to nuclear energy - only with a much bigger boom.
Long term trips would require advances in waste disposal/recycling technologies, probably new types of engineered food - made to grow fast in poor conditions, and be highly nutritious. Landing (and staying) on a surface would require new materials, light, strong - probably advances in bucky tube [http://en.wikipedia.org/wiki/Bucky_tube] technology. All of which would have real benefits on the home front.
Even long-term research into terraforming other planets (Mars?), would probably give us insights into how atmospheres are structured, allowing us a better opportunity to reverse the damage we've done to ours.
As for running into aliens. Well, I'm on the fence about that one. Statistically, there have to be other sentient aliens out there. And mathematically, we should have been aware of them by now, at some point in our history. In fact, it's strange that we haven't if you consider the Fermi Paradox [http://en.wikipedia.org/wiki/Fermi_Paradox] legitimate. It's possible that at this point in the timeline of the Universe, we're the only technologically capable civilization in this galactic arm, or galaxy as a whole. But certainly there have been some before us, and will be after us.
But we should definitely get out there and looking around ASAP. Earth is delicate, and not just the ecosystem. It's a small planet in a big Universe, and we should clear the fog around us as much as possible. Heck, 100 years ago, maybe a star went nova in a nearby star system. Maybe that caused the orbits of that stars planets to go all whacky, and some of them lost their moons. Perhaps, one of those moons swung into to the newly collapsed neutron star, and narrowly missed it - accelerating the rogue moon to a near-c velocity, that just happens to collide with our Earth orbit in the next 10 years or so. By staying on Earth, we'd never see it coming. At that speed, and inbound trajectory, there would be little to no warning, and no opportunity to prepare. It would crack our planet in two in the blink of an eye.
We should definitely push forward and outward. The dinosaur extinction was not a one-time event.