Why would the crew of the Prometheus crew get a brief on their mission and location only after they arrived? It would seem to me that if I was going into stasis I'd want to know where I'd wake up and what I'd be doing there... Was the reasoning behind this explained or hinted at in the movie?
From an in-universe perspective;
It seems like this is the norm for operations at the time. Crew members even place bets as to what sort of mission they are going to be briefed on. These missions are very expensive as discussed in the film, and this one in particular was very secretive. I feel like it is typical for people to not be told all of the details for secretive missions until arrival (and/or a certain time when the mission can't be compromised).
It is likely that the crew were given at least small tidbits of information, but remember that this is a relatively early point in their space faring. The information they would likely care most about would be; the amount of time you will be there, the relative danger involved, and the amount you would be getting paid. They wouldn't really care about "where" they were going as it was all relatively unexplored anyways, and if they had been told why they were going before they set off, there would be the potential for the mission to be compromised. (In the film you can see that not everyone enjoyed the concept of being created by someone other than God.)
Remember too that it is a company that is sending them, not the government. There would certainly be some competition from rival companies, or perhaps the government itself, if they knew that Weyland Industries were taking on a mission to find and speak to the Engineers.
From an out-of-universe perspective;
It sped the movie along in my opinion. Basically, it saved them from having to develop each characters' motivations for wanting to go, and allowed the characters to have varying reactions to the situations presented them. If everyone were told before hand, then it would be safe to assume that those chosen to go would be the people who actually wanted to go, and you wouldn't get the same sort of insubordinate or conniving behavior. We are presented with a ship full of crew members that have varying levels of involvement, interest, and intrigue, instead of a unified group of people all working towards the same goal.
For a classified mission the crew members would not be told anymore than they absolutely need to know. As crew members in that kind of organisation this would be familiar and not at all weird, this is shown by them betting.
If they were briefed before leaving the chance for a security breach increases massively and unnecessarily.
If they had explained it before, the people watching the movie wouldn't know what the hell they were doing on that planet. The explanations were necessary for the public to understand what was going on. It struck me as pretty stupid as well, but it's the same as when an evil guy explains his evil plan to a good guy before killing him.