Wow, it's the first time I've played Team Fortress 2, and I must say it's awesome! I haven't had this much fun playing an online FPS game for a long time. Screw CoD4 and Halo 3, this is the multiplayer game to play!

Glad you like it, it really is one of the best multiplayer games ever.  I have so much fun playing it everytime I do.  The next time you are playing arena mode, try to get your team to rush the same class.  Now that is not only fun but hilarious at the other teams reaction to having like 10 pyros or heavys coming at them.

Well, I've only tried the Heavy and Soldier class, but I think I'll try Engineer and Demoman next time I play. Good idea though!

I've played in a Mario Kart arena (mod), and I thought the guy did a really good job with it. I wonder if the normal maps are more satisfying than modded ones?

I have only played stock arena maps so far. I usually really enjoy custom maps because its a good change of pace. I guess people are already making custom maps.
I will try to try some later.

Well, I just got in an achievement server about 20 minutes ago, and it was chaotic! I tried all the different classes, but I still don't know how to build stuff with the Engineer...

Press 4 and you get a building menu. You have to have enough metal. The sentry gun can be upgraded twice from its original level. The teleporters are self explanatory and the dispenser gives out health, ammo and metal but only 40 at a time.
Press 5 and you get the destruction menu where you can blow up your stuff to replace it in a better spot and other reasons.

downloading now, been meaning to check TF2 out for a while

To expand on what Quarashi was saying:

When standing where you want to build press 4 to bring up the build menu.  On the build menu 1=Sentry Gun(130 metal), 2=Dispenser(100 metal), 3=Teleporter Entrance (125), and 4=Teleporter Exit(125). Note that you can turn your structures 90 degrees by right clicking when you are about to place them.

You can speed up the building of your buildings by pulling out the wrench and beating on them like you would an enemy.  You can also upgrade your sentry the same way but to upgrade it cost a total of 200 metal per level (it takes 25 metal away from you per hit with the wrench).  The first sentry level is slow firing and slows enemies a bit when hit.  The second level is double barrelled minigun fun and fires much faster with even more slow for the enemy when hit.  The third level adds a quad-rocket launcher that delivers a can of whoop-ass to your slowed enemies.

Good strategies as an engineer are usually to set up your Dispenser first at the beginning of a map because it provides you with extra metal and health to speed up the building of other structures.  Sometimes if you are in a far forward position trying to build you may need to go straight to the Sentry Gun but hopefully if you do that you are building right on or near an ammo pickup to help you out.

You will also need to be careful of spies as they can sap your buildings rendering them ineffective until you repair them (ie hit them with the wrench 2 times).  A good method to ward off spies is to find a position for yourself that only leaves a narrow path of approach and smack anyone who comes near you with a wrench a couple of times.  A few other good hints are that if you can't walk through the person they are a spy.  Damage isn't always readily apparent on enemy spies so if you're not convinced keep checking.

Oh and a key part of being an NG is ducking behind your Sentry when its under fire and repeatedly whacking it with a wrench to heal it so it has extra time (and ammo) to kill your enemies.

^He said it perfectly. Also a good way to spycheck is when you can't run through someone. You can usually run through people on your team unless theres some type of mod in the server, but thats a dead give away when spy checking. Also check the names of people because if someone comes running out you with your name, I will let you figure that out yourself.