I was in the same situation as you when I got my xbox 360 on Christmas of 2007. I had never played Halo 1 or 2, but I decided to buy Halo 3. In my experience, I can say that yes, there will be things that you might not understand just by playing this game alone, but the fun far outweighs that. In other words, just because you might not understand the story is no reason to not buy this game, it's a great game regardless. I played it and loved it, and to this day I still haven't played Halo 1 or 2.
The key aspect is the online multiplayer, it's great. But with so many new games out, there might not be as many people playing Halo 3 online as there used to be a few short months ago. So I suggest that you look into Gears of War 2 or Call of Duty World at War, since the releases of these games made a lot of people stop playing Halo 3. Both are great games that you are sure to enjoy if you like Halo 3.