Excellent question. Mars is the obvious answer, but maybe not the only one.
In ST 2063 is the year warp drive is invented, but I don't see any evidence that this fiction
will ever become fact. I don't know if any of the moons out there (of Jupiter, Saturn etc) are any better than Mars.
Mars is cold, but it is the only logical choice, as I see it.
I would be very interested if anyone has any specific ideas of places other than Mars.
Our moon doesn't have the most friendly climate. There is the idea of hollowed out asteroids, which could be used like space ships. (aka planetoids)
As for the how, well right now we don't have the ability to go to other solar systems, but exploring this one is not really a question of "how", but of the willingness to spend an awful lot of money.
There's Titan (which has an atmosphere and a magnetic field, to protect against radiation), Europa (which likely has a moon-wide ocean of liquid water underneath its icy crust), there's the Moon (now that we know it has water ice), and the newly-discovered planet in the Goldilocks habitable zone with probably liquid water which orbits Proxima Cenaturi our closest neighboring star.
