Luna has an approximate mass of 7.36x10^22 kg. Earth's is 5.974x10^24 kg. That means, at the distance between them, the force of gravitation is 1.98 x 10^20 N. This creates the tidal effect, known today.

A moon base would not remove any mass from the moon, so gravitational force would remain nearly the same. (Save the added mass of the population)

The total volume of an O'Neill Cylinder is 1875km^3, or so. At a 50% pressure atmosphere, made of 2meter-thick steel walls, assuming 3.29 x 10^13 kg mass for each mirror, the total mass of an O'Neill is 1.26 x 10^17 kg. This is easily the largest structure ever created by mankind, and most definitely the heaviest. Making it entirely out of Lunar material puts the remaining mass of Luna at 7.359 x 10^22kg. In other words, constructing it used a fraction of a percent of Luna's mass, so small, I'm too tired to calculate it. This has the obvious implication that gravity, thus tides, are unaffected by even the most massive construction projects undertaken by mankind. I won't even calculate, right now, at what point an even noticeable difference in tides would begin, nor how much we'd have to scrape out of the sacred white light in the sky.

Humanity will be extinct before we see the tides change due to the moon's migration away.

Faster-than-light has yet to be proven to even exist by the general laws of physics. It's hundreds of years before we can even achieve notable percentages of the speed of light. You can't even consider extrasolar colonization at this stage. You can hardly consider exploration, because, we simply do not know how to restore a body from cryogenic sleep. Warp doesn't seem remotely possible at this stage, and I refuse to get into why right now.

To colonize a planet with equal gravity to Earth, but with a different atmosphere, essentially requires building a series of small space colonies on the surface. Even then, you have to find the planets, and go there. It gets trickier if gravity is different.

Right now, our best bet at putting populations out of the cradle, is to put them in colonies in orbit.