The concept of Team Fortress really intrigues me, but sadly my laptop is ridiculously below average and incapable of running TF2. I own TF Classic as a result of purchasing a Half-Life game pack last year, and it runs just fine, so I was thinking of giving it a go sometime. However, given how old this game is, I'm guessing its userbase is pretty devoted, and therefore pretty experienced, and that the game isn't particularly geared towards new players. Does anybody here on Giant Bomb play this game, and if so, do you have any tips for a newcomer to the franchise who wants to get started?

You're right, the userbase is pretty devoted. When I started playing this competitively (PCWs, pickups), I had been playing fast paced FPS games for a long time (Quake 3, Painkiller, Natural-Selection etc.), and very good control over the GoldSrc engine's (Half-Life engine) controls and movement. 
I would recommend playing some conc_ maps to get a grip of the movement, since concing plays a very big role in the offensive competitive gameplay of TFC. This game has a very steep learning curve. 
When you feel you have a good hold of the game go to #tfc.pickup and !add up to join a pickup. As in all games, the small community always seems friendly apart from the usual newbhater. 
 
Good luck :)