As for the destination, the moon and Mars are the obvious choices, but what else would make sense?

For our first colony, the Moon or Mars makes sense. The Moon would be the easier choice, while Mars would be much farther away and perhaps be more interesting for that and other reasons. Ganymede or the other two big icy moons of Jupiter would be interesting too. Or maybe the first mission to establish a mining operation on an asteroid?

How long would it take to get there?

Depends. The Moon, a few days. Mars, with current technology around six months, although that could be reduced if you use plasma propulsion for example.

What could be the goals of the mission?

Maybe a precursor/pathfinder mission to prepare for the following big colony missions? Or maybe just a research base?

Any events or tasks that could punctuate an otherwise predictably boring long trip?

Just from our space missions there are a number of problems that has occurred in reality. Problems with heating or cooling, explosions, fires, collisions between ships, problems with carbon-dioxide scrubbing, loss of attitude control... The problems alone that can come up are many, if you add in the danger of space flight, the human factor, computer hardware/software failures. Also, radiation, micrometeorites, the effect of low G on the human body, the possible effects psychologically of living for months in a confined space far away from home, etc. Tasks could include course alterations, scientific experiments and observations, daily maintenance, etc.

Any possible sightseeing for beautiful VFX shots?

On Mars of course you have the Valles Marineris, Olympus Mons, dried up floodbeds, craters, volcanoes, dust deviles, the annual dust storm that pretty much covers up the entire planet... etc. If you establish a base on one of Mars's two moons, Mars itself would be prominent and probably a beautiful sight in the sky. On the Moon, all kinds of interesting things. Mountains, craters, maria, etc.

What would be the crew?

I'm thinking international, no matter if it's a government project or entirely private endeavour. Men and women, of course. For the establishment of a research base, I would have only adults. Could be different on a larger colony mission of course. Artificial intelligence would be part of it, whether it's the ship's computer, robots or intelligent systems built-in in the spacesuits for example.

AVGFree anyone?

A scientific Lunar base will be a place of exploration and science. It can be a national flagship or whatever, but you really should not say that it can't therefore do real science.