I'd say start as scout, that's how I started. He's one of the harder classes to play, but he's uncomplicated and the team doesn't rely on him doing certain things at certain times. The sniper is the same, if you like sniping.
The pyro, heavy, soldier and demo are also fairly straightforward fighting guys (the demo perhaps takes a little more getting used to). The downside of those for a beginner is that sometimes a medic will decide to uber you, and then suddenly all responsibility is on you. That's fine once you know what to do, but when you're still learning the maps it can be a thing.
Oh yeah, one word of warning about the scout: On certain maps (mainly Goldrush and Dustbowl, scout straight up does not work. There's not enough space for him to run around, and not enough routes for him to sneak up behind them. RED builds a defence and BLU tries to break in, and there's nowhere for a scout to find the 1 on 1 fights he likes. By the way, the scout prefers to pick people off on their own.
Avoid the main battle line, try to find people on the flanks, and don't worry about killing them as soon as you see them. Use your speed to attack from different angles and monopolise health kits; as long as they can't heal you have as much time as you need to bring them down. Oh, and always kill the medic first. You can't keep up enough sustained damage under fire to kill someone through the healing. Don't get hit, don't let them heal. You can kill people fast, but only if you catch them on the back foot. And if you meet a scout, you can't run away so it's a fight to the death.
