I'm a member of the stardestroyer.net forums, and there are some very sharp people on that site who would be happy to give you a hand with the technical side of things. They also have a user-fiction section just for writing stories, and some of the ones posted there are pretty damn good.
Just be polite. And make sure you have a thick skin. And do your homework first.
As for your questions, I can take a stab at them...
As for the destination, the moon and Mars are the obvious choices, but what else would make sense?
Near Earth Asteroids, Venus (reasonably habitable 50 km up: http://en.wikipedia.org/wiki/Colonization_of_Venus#Aerostat_habitats_and_floating_cities), Phobos, Deimos. Moons of Saturn might work, Titan and Enceladus being the more interesting ones.
How long would it take to get there?
Also the rest of the atomic rockets site I just linked has very good stuff for just this type of question.
What could be the goals of the mission?
There are maybe three broad categories, I'd say: political (Why did we go to the moon anyway? To rub it in the Soviet's faces.), monetary, and scientific.
Beyond that, well, you tell me, you're the writer. He3 mining on the moon? Political/Religious refugees? Life found on Mars means everyone wants to go see it? There are a lot of semi-plausible explanations. Which is all you need to start a rattling good yarn (sometimes not even then).
Any events or tasks that could punctuate an otherwise predictably boring long trip?
Micrometeroid punches a hole in the ship. Solar CME event burps a lot of radiation at the ship. The engine stops working. The AE-35 communication dish develops a fault and they can't talk to earth. The plants (the ones that provide air and feed people, you know) get sick/die. The biologist comes unglued and murders someone. I mean, this is stuff off the top of my head, man.
And there's always turnaround day for continuous acceleration ships. (The fastest way to get anywhere in space besides FTL travel is a continuous acceleration route, where you burn the engine to speed up halfway to your destination, then flip the ship over and burn the engine to slow down. Flipping the ship you have to do with the engine off, so everyone goes weightless for a few hours or a day while the ship turns end for end.) In some universes this is traditionally accompanied by a celebration or a special dinner or something, along with funny things like (say) bolting the floor furniture to the ceiling or having the most junior officer head up dinner instead of the captain.
Any possible sightseeing for beautiful VFX shots?
Space is beautiful, kid. There are always good VFX shots.
What would be the crew?
Captain, doctor, science, communications, pilot, engineering (the astute among you will notice I'm actually listing off bridge positions from the original Star Trek...)
Ok, come on, kid, I'm not going to do all your homework for you. If you can't even be bothered to look up or think up common crew positions, why bother helping those who won't help themselves?
Seriously, most of this stuff could be answered with some intelligent usage of Wikipedia and Google and a few hours of spare time. I answered this because I was bored and was familiar with it, but if you actually care, why aren't you looking this up for yourself? If you did already, say that you did, but want geek's valuable opinions. (and they'll fall all over themselves to give it).
Because right now the summary looks like you are lazy and can't be arsed to look this stuff up yourself. Do your homework, and intelligent people will be much more interested in helping you help yourself.