A couple interesting effects that were not mentioned in previous answers:
There would probably be a significant group of people who deny the existence of aliens even in the face of evidence. Just look at how many people do not believe we actually landed on the moon. There may even be "denial" cults that form.
One thing that would immediately happen which most fiction does NOT mention is the instant technological leap that would occur as soon as humanity KNEW for a fact that faster than light travel IS possible. As soon as you know something can happen, it becomes a lot easier to develop it. Who knows what the mere theoretical knowledge of FTL travel might do to telecoms? How does FTL interact with time (as in, if we plot out a curve base don Einstein's relativity theory, we find that time becomes a relative factor at different speeds; hence, why not time travel?)
Just looking at the design of the ship in a telescope would give Earth engineers tons of ideas. Power sources, structural design, etc. After WWII, German designs for high speed jet fighters revolutionized US and Soviet designs just by looking at how they had tilted the wings back at an angle to stabilize the craft at near-mach speeds. Who knows what you might learn by looking at the physical design of a huge FTL spaceship. This obviously doesn't count the incredible advance in metallurgy/materials technology that we could get by doing nothing more than hitting the surface of the ship with a spectroscopic analysis (which I believe can be done from Earth, as we might do to the surface of the moon using telescopes and lasers).
There would be cultural phenomenon. Fads, trends. After airplanes were made reasonably common, they became this overwhelming fad in the 1920s. You had airplanes in wallpaper designs, stylized airplanes zooming on magazine covers, etc etc etc. People were crazy for anything aerodynamic and with wings. In the 1950s, we went nuts for rockets and fins because we had just orbited the first man around the planet. I could see anything related to the aliens creep into music videos, movies, games, etc etc. It would be a global obsession for a good decade or so at least.
Politically, it would be chaos. Politicians who led countries (most of which, around the world, remember, are actually tyrants who can do pretty much whatever they want) would scramble to get an advantage. Every single one of them would be desperate to communicate first and cut a separate deal with the aliens before anyone else in hopes that they get supertech their rivals don't have. This would be such an overwhelming need (remember, even if YOU don't want to do this but you live anywhere near someone who does, you HAVE to in order to defend yourself) that there would be a HUGE space race to GET TO THE ALIENS before everybody else. We would not wait for them to get here, we would CHASE them from across the solar system where they would be bombarded by 80 different petty dictators in barely-bolted together space shuttles trying to communicate (and get a better deal first).
I predict that given the state of geopolitics today, SEVERAL small, brushfire wars would break out as mid-sized nations tried to sabotage one another's rushed space efforts.