Hey, glad to see you enjoyed it. Yeah, TF2 is a great game to get acquainted with the FPS genre. If you have any questions feel free to ask me. Below I will give some tips:
---Classes:
Scout: Do NOT play on servers with more than 20 people. Also, be aware of ranges, your scattergun (the primary shotgun) is not good for distances over about 20 feet. For long range, switch to your secondary weapon, the pistol
Soldier: Aim at peoples feet, not their heads. The rockets cause splash damage and so aiming at their feet will hurt them, but you may miss and cause no harm if you aim at their head.
Pyro: As you said, this is a teamwork game, and the Pyro is a good example of that. Your flamethrower will set people alight, and the afterburn causes 50 damage. This is where you leave the fray, so that another class with more killing power can finish the job. Basically, hide, ambush, set everyone alight, run while they burn. If they are beyond the range of your flamethrower, switch to shotgun.
Demoman: You are the only character with grenades, use them wisely! What I usually do is shoot them around corners or like artillery (over obstacles, etc.). Your secondary weapon shoots sticky-grenades. Shoot some of these onto a control point or cart (depending on the map) and right-click when an enemy steps on them. You can have 8 lying around at any one time.
Heavy: Shoot everything you see! Right click to spin your barrel without shooting (useful when going around corners).
Engineer: You are weak. You have little firepower. However, you compensate by being able to build 4 buildings. Build teleports to bring your allies to the front line. Build dispensers to provide your team with health and ammo (and you with metal to build buildings). Lastly, build sentry guns to defend the area. Beware, sentries have a limited range. Fix buildings by hitting them with your wrench. The sentry can be upgraded first to gatteling guns, then by adding rockets. It costs 200 metal per upgrade. I really don't recommend this class for a beginner.
Sniper: Right click to zoom, then aim at peoples heads or chests and fire away! Beware, the middle of your crosshair emits a laser, which is visible to the enemy team and will give you away. Hide it by pointing it off to one side (example: if aiming through a doorway, aim at the frame and only move to take your shot when someone appears).
Medic: As soon as you spawn, switch to your secondary weapon. Aim at your team mates and left click. They will be healed. Give priority to the ones asking for assistance (they have a speech bubble with a red cross above their heads). You can "over-heal" people to 150% of their normal health. Only use your primary weapon if no one else is around and someone is bothering you. Beware, it shoots in an arc, not straight like other weapons. After healing for a while, you will have charged your "uber-charge." aim at a heavy, demo, soldier, or pyro (on occasion), and right-click.
Spy: Do NOT play this class. It is way too complicated. Even I am still trying to master it.
---Terms:
Spawn: The place where you appear.
Uber-Charge: if you ever see an ally or enemy that is glowing blue or red (depending on team), this is an uber-charge (this is what the Swedish guy was talking about). The uber-charge means for 10 seconds you are invincible. People use this to charge forth into enemy lines and wreak havoc.
Critical Hits, or "Crits": Occasionally, you will fire "critical" ordinance along with your normal ammo. These do a lot more damage than normal, and usually kill in the cases of a crit-rocket or grenade. They are characterized by glowing red or blue (depending on team) and emit a lightning sound when fired. The exception are critical melee hits. These don't glow, they just hurt!
Taunts: Pressing 'g' will cause you to do a weapon-specific taunt. It is usually a bad idea to do mid-battle as you lose control of your character for a few seconds.
Melee: Your third weapon in most classes this only affects the people adjacent to you.
Point: those circles with the red or blue hologram in the middle. These are the crux of "control" maps. By standing on an enemy or neutral CP, you can convert it to your own.
Cart: in "payload" maps, the cart in the track with the 1940's-style atomic bomb in it. By standing next to it if you are blue team, you can push it forward. The goal is to get it to the other side of the map
Intelligence: in "Capture the flag" maps, its the briefcase with the papers falling out. In your control room there is this briefcase, the enemy will try to come and get it, and you should go and try to get theirs. The point is to grab it and bring it back to your own intelligence room alive. If you die, you drop it where you died and it then counts down for a minute or two. If no one picks it up before it runs out of time, it returns to its original location.
---Game-Related:
Teams: It is a typical Red vs. Blue game. The two teams represent two warring corporations that are involved in demolitions, mining, construction, etc. and this is reflected in the maps. Blue is the attacking team, Red is the defending.
Maps:
-Gravelpit: Diamond shaped map. Blue starts at the bottom, red starts at the top and controls the two side points. The point here is for blue to take the two side points and then converge on the top. Red's goal is to hold out until the time is out.
-Hydro: Totally random in almost every way. If you listen to the commentary, this is intentional.
-2Fort: Red and Blue each own a fort separated by a common moat and bridge. This is s "capture the flag" map.
-Dustbowl: This is actually 4 different maps all combined into one. You only play one section at a time. The different sections are separated by walls and doors that only open to one map at a time, and as such, if you look through the fences you can see other areas of the map that you can play, just not during that round. This is the typical "control" map.
-Goldrush: This is like Dustbowl; 4 maps in one. Except here it is a "payload" map, not a "control" map.
Hotkeys (or at least how I set them, I forget what default is, its pretty close):
-W: Foward
-A: Strafe left
-S: Backwards
-D: Strafe right
-E: Call for medic
-R: reload
-G: taunt
-T: Spray your spray, upload one in "options"
-Y: All speak (both teams can read what you say)
-U: Team speak (only your team sees it)
-V: Voice (if you have a mic, it is recommended you do)
-CTRL: Crouch
-SPACE: Jump
-ENTER: sent what you typed if you opened up chat
-TAB: Check player stats, like their score, ranking, and what class they are if they are on your team
-Z, X, C: pre-coded vocal commands, such as "help!" "sentry ahead!" or "Thanks!"
-Left-click: fire primary weapon
-Right-click: alternative command, such as activate uber-charge or zoom in on sniper rifle
-Scroll Wheel: Alternate between weapons, left click to make your selection
-Q: Switch between your current weapon and your last. If you only have had one weapon out, this switches to your secondary and back.
-1-5: switch between weapons 1-5. Most classes only have 3, but the engineer has 5 and the spy has 4.
Well I think that about covers it. Again, feel free to PM me if you have any questions. Happy Hunting!
PS: just a little trivia, each character has a unique accent.
Scout: Boston
Soldier: Standard American Sergent (read: Full Metal Jacket)
Pyro: No one really knows... My personal guess is he is from the West Coast, but that's just me
Demoman: Even though he is black, he's Scottish
Heavy: Russian brute
Engineer: Texan
Medic: German
Sniper: Australian
Spy: South English