I'm starting a new character, but still don't know much about the game itself. I would like a good character for battlegrounds and dungeons. I really enjoy short-ranged attacking, but don't like being a tank.

Just play the game and get a feel for it, most classes are pretty decent and will hold their own. I have seen Druids, Rogues, Warlocks and yes, even Warriors do great DPS in the game.

One of the things to remember is "threat management" if you think you are generating too much threat on a target either change targets, or if you are fighting a boss, you'll either need to use a spell or skill to remove that threat or back off for second.

With a really good Tank (or Tanks) you shouldn't have to worry too much about your threat management, but sometimes you'll need to consider it. If you are playing in battlegrounds, it isn't a issue and I wouldn't worry about it, 9 out of 10 times battlegrounds is all about the gear you tote, not your choice in classes.

My favorite is a Death Knight.

I'd recommend Shaman. They have a melee tree (Enhancement) built for dual wielding. It's been a while since I've played but they might still have an aggro-reduction talent in there as well.

You can also run a rogue or cat Druid (And only cat druid, as the other trees are ranged and tank), who's entire set is based on melee. The other melee classes (Paladin and Warrior) are lacking in aggro reduction as well as a number of their attacks having increased aggro production.

So it's more of a playstyle thing. Enhancement Shaman get in there and tear it up like a fighter without the tanking abilities (Plus shocks are good for spellcasters and runners), and rogues/cats tend to load damage in bursts with combos (Have to be careful when dealing with a new/undergeared/bad tank).