As far as I'm aware, space is not infinite, but it's still expanding from the big bang, and it's expanding quick enough for it to be practically infinite, as we cant reach the edges.
We know it's expanding due to the Doppler effect. As you know, sound travels in waves, and the pitch of a noise depends on the frequency. Imagine an ambulance driving towards you. It emits a sound, which travels to you, and you hear the noise. Then it emits the sound again, but because its travelling towards you, its a bit closer. As such, it arrives a bit sooner than the last one, and the frequency appears to increase. As such, it sounds a higher pitch. The same thing happens when its travelling away from you, except in reverse. This is the Doppler effect.
As light is also a wave, it also experiences the Doppler effect, except with light, changing frequencies changes the colour. A low frequency gives red. By examining far off galaxies, we have noticed a very distinct red tint, showing us that they are accelerating away from us, and fast.
As for living on another planet, I really doubt it. Astronauts can only stay in space for a few months, before they start experiencing major problems due to the lack of gravity, and to get anywhere habitable would take us many, many years. About the closest thing I've thought of is getting a load of eggs and sperm, loading them onto a ship, and sending them off. Then when they reach the planet, the sperm and eggs are mixed, and then the babies are looked after by robots. And even then, the idea is ridiculous.
Unless of course we master wormholes, but I know very little about that