The Smeagol flashback was rather nice, IMO, except that part where he bites the fish. Creepy.
I think the main reason PJ added in that Smeagol flashback was to emphasize the pity one should feel for him. Except for a brief talk between Frodo and Gollum in TT, Gollum is just seen as a nasty, slimy creature to non-book readers. Gollum's flashback makes it clear how the Ring took control of him, and that he deserves some pity. I especially like how they showed the current Gollum right after the flashback, to emphasize this.
The Fellowship prologue was my favorite though. Starting off with a huge battle scene lets people know that this isn't a boring movie you're about to watch. And PJ only shows you a glimpse of the battle, so as to not spoil the "real" battles.
Gandalf's fight with the balrog was also great, in that it
continued from a previously-seen scene. But it had a major flaw, which stops it from being my favorite flashback. It practically gives away that Gandalf's still alive right at the beginning of the movie. The only thing people have to wonder is: "How did he survive and when/how will he return?"