What should I consider when selecting a Prize to take with my Tournament?
A fair amount of the time, Followers is going to be the strongest prize. If the board doesn't have another way to give Curses, and you are getting to choose a prize relatively quickly, giving out the only Curses in the game should have a pretty big impact. Also, bringing your opponent down to a 3-card hand will make it more difficult for him to buy those important Provinces.

After that, Trusty Steed tends to be quite strong. Be sure to think about all the options that it offers; it's quite common to just choose +2 Cards +2 Actions, but that in itself isn't all that great... it's a level 2 City, and Cities are only good when you get a lot of them. But all the options have their place.

All the others are more situational:

Princess can be necessary if there's no other +Buy card.
Trusty Steed can be necessary if there's no other +2 Actions card.

I don't think I'd ever take Bag of Gold or Diadem as the first prize. But either of them can be good if other prizes are already gone. Bag of Gold is good mostly in a game towards the end when you need a little more boost to get the last or last couple Provinces. I rarely see Diadem worth more than 2 or 3, and when it is, you probably already have a deck with enough money anyway. However, I'll often choose it because I need that extra 2 or 3 this turn.