Not the very first quests, those can only be done by Belfs (unless they finally fixed that). Other races can do the quests that start with the ruins of Silvermoon though. I don't care for belfs much myself though, I only go out there for the recipes and rep.
I agree with Mulgore, the Tauren zone. I love my moo cows, and I love the area, but it's sooooooooooo spread out, and a lot of things are up a hill, which can make targetting harder.
I say try whatever catches your attention. I started with a trial too, and trust me, 10 days will be plenty of time to try a few races -- plus, trolls and Orcs share the same zone, as do gnomes and dwarves. If you do one race, there won't be any difference when doing the other race, other than the butt you're looking at and the classes that are available. And the tiny viewpoint from gnomes :ph34r:
And holler when you pick a realm, one of us might be on there too :wave: