Um, having done the trial:
As I understand it, WG (Wintergrasp) battleground and Grizzly Hills are both high level places (70-80), so I don't think you'll have to worry about them. Not sure that low level characters can do anything there now...plus they require Wrath of the Lich King, which you probably don't have on that trial.
Pick any server, it won't matter a whole lot during your 20 levels of a 10-day trial. Seriously. Just get a feel for the different races and classes, see which ones you like and which you don't, and try out the normal and RPG servers. You can make up to nine characters on an account, I forget how many servers you can play on but it's limited as well -- but not so limited that you can't have a few dozen alts scattered around, which will be way too many during the trial period. IMO a PvP server should be chosen after you decide to get the game, because after level 20 is when you'll be entering shared zones and actually encounter PvP situations, at least those in which you'll have a real chance of surviving. The RPG servers are a little odd as well, because some people are more into roleplaying than others. I play on two RPG realms, but never really see much of the hardcoe roleplaying (like using certain language or walking everywhere instead of running), just people getting into their characters and the game lore a bit more than on PvE servers. Proudmoore and Sisters of Elune have both been pretty good so far.
You also can't join a guild on a trial account, but you can pay attention to who you're seeing often. Some guilds are just for hanging out, others are for leveling, and some have specific requirements, like those that are just for higher-level characters who do lots of raids (a type of dungeon). Being in a guild can have benefits, if it's a good guild and you get along with the members.
One important thing to know if you use more than one email account to make other trials: once the 10 days is up, you can't access those characters again until you register the account (i.e. buy the game), and I never did find a way to link the emails together -- not even sure if it's permitted, to be honest. That means you'll lose some of your characters, so be sure they aren't the ones you want to keep if you end up getting the game.
According to their FAQ, both of you do need to be on at the same time in order for the 3x to apply, plus you must both be grouped together and remain near each other (so grouping while on different continents won't count). Shouldn't be hard though if you're playing together, just be sure you turn in quests together too so you can get 3x experience there as well (a VERY nice perk). And really, between level 20 being fairly easy to reach, getting 2x experience from logging out for a while in an inn or major city (as long as the experience bar is blue instead of purple), and 3x RaF experience, you'll easily hit 20 within just a few days, if you play a couple hours each day. There's a LOT to see and do in WoW, and leveling up too fast can mean missing a lot of it, so don't worry too much about hitting 80 as fast as possible. The destination is nice, but the journey can be too ^_^
Also, you'll eventually learn about the auction house (AH), and if it's something that interests you and you think you'll want to use the AH when you buy the game, save things in the bank so you'll have them when that character's account is activated. It can be a big help in getting your character "officially" started