Scout: Use your jump to escape large groups. A lot of times, you can use it to get onto scout- and soldier- only paths, like the top of the 2fort bridge. Mad Milk is good in a pinch; throw it then shoot your enemy to heal. If you know how to use it, use the sandman, but personally, I love running around as a Mackerel scout for the humiliation factor. If you're good at dodging, that's an awesome way to play, running around enemies and beating them with a fish.
Soldier: Use the blast radius to your advantage. There's a reason I broke down my Direct Hit for scrap, direct hits are really hard to hit. The buff banner is useful; even if you only get the assist, you get rage, so you don't have to aim for kills if there's a lot of your team around (that can be really helpful if you're not too good at soldier kills.)
Demo: Use rebounds off of walls. A lot of my kills have been by bouncing bombs around corners to surprise enemies who thought they'd avoided me. Consider running demoknight; the health and damage bonus for kills is really useful.
Pyro: Now this is mine. Personally, I use a Strange vanilla Flamethrower. It's got good damage and no airblast cost penalty. At close ranges, this is what you want to use; another option is to go Degreaser/Axetinguisher within melee range. The only disadvantage with that is that airblasts take a huge amount of ammo. If you get in trouble against non-pyro enemies, just run; often the burn will kill them. Airblast against soldiers, demos, and (less known) huntsman snipers; if I remember right, hits with reflected projectiles are mini-crits, too. I use the Flare Gun for a secondary. After a while, the funky arc is easy to project, you just need to use it enough to figure it out. If you can lead your target, it's one of the most deadly weapons out there. Definitely one of the most underrated weapons out there. The damage isn't great, but the burn is long, and it crits burning enemies. I've helped hold the second point on the 2nd stage of Dustbowl (and been one of the best players on my team) by having a dispenser on one side of the bridge-thing and just popping out, taking quick aim, and hitting enemies coming out of the tunnels, then hiding back at the dispenser, going to the right and looking up occasionally. As for melee weapons, I haven't really experimented with them, because I usually use my flamethrower at melee range.