Soldier, is likely my best class. I guess to deal with each area individually first.
Class wise, Your biggest, and often toughest targets to deal with will be based on where you are fighting. A scout has you at a disadvantage in an open area usually, just as a demo or heavy will fighting where you can't take advantage of cover. Learn weapon sounds and how they look, as the pyro can counter your best manner of attacks in a direct manner (the airblast) or leave you wondering how you died (Backburner) With heavy, its wether to engage from long range, or to pepper around a corner. Natascha really hurts you just the same as sasha (though beat a quick retreat to any cover with any crits either way, distance doesnt reduce the damage they do.) If the heavy has a medic with him (many will, often buddies), you want to stop his life line, and to do that, you have one of the best tools long or shot range to do so. If you are at long range, your aiming for your 3rd/4th rockets to hit and make him run taking his heavy with him. (or to use that uber early, both help you and your team) The first shot is towards him, aiming to the ground, looking at where your firing, guess where he will run to, and time that 2nd immediate fire so that he either stays put, bunkers behind the heavy, or runs towars rocket #3 and #4. If he bunkers, then you have an easy target aiming behind the heavy.
They pyro, sadly, or not, depending on which class you ask has a good counter to your rockets, though not all users are super at using the anti-rocket weapon. The two big things that favor you in the fight, is the cost of the airblast (at 25 ammo, its very pricey) and its refire rate. You can fire faster than the reflect is, maybe 1-2 rockets of the four can be bounced back depending on distance. (longer distances mean more reflects if you space the firing) Another is where you hit the floor on a pyro, if you aim before, even if you use more rockets, you can stop his forward momentum (instead of hitting the back and sending him flying flamer first into you.)
Medics, normally, are not an issue, exceptions being when you have the charge after killing their buddy and are low health/ammo. Crits have changed, with ranged maxing out at 4% i think, but melee is unchanged, so that ubersaw/bone saw, can easily have a near 30% (melee carries a double crit chance) The advantage to you, is having killed somone, if you can connect with your shovel (if you choose to) All it will take is one hit to down him. The other alternitive is the ever trusted shotgun, at close range, one, maybe 2 good hits will take him down without his blutsauger peppering you.
Demomen, fall into two varieties, the first being the sticky bomb men. They lack understanding of what a grenade is, and will use the sticky ad-hocly as a ghetto rocketlauncher, usually, depending on where, you want to shove him back into a corner and either use that shotgun, or juggle him with a rocket/shotgun combo as once airborne, he often is at odds with damage given the recent nerf. (it takes about 5 seconds from when it hits the ground to start upping on the damage) You can, situation, permitting, try taking to the skies in a jump and bombarding from above, but your aiming at a long rjump, not air time, so he can't land stickies at your spot to reduce you to soldier mustard. The Piping scotsmen, is the other, and will often have traps waiting here and there, but use the grenade to nail you from close range, or odd angles. And can trick you into walking over missed grenades. Situational awareness, helps avoid them, and often circle around a corner first, rather than going straight through, to avoid a set sticky trap, to help nail him.
Scouts, can be a love or hate affair, depending on what they use. The sandman nerf (thank god) makes you take half damage, but with fan, you can expect about a 50-75hp hit after getting stunned with one, if your in the open, taking a sniper round too if they are around. Best bet, if you see a sandman scout, is the juggle with the rocket, is a death blow to them, once you send them up, send a second rocket, or shotgun if your out to drop them. Bonk, you will mostly see them trying to use running away or by you to get to health/ammo, and just physically blocking them, with rocket/shotgun makes them retreat if your in a narrow spot, but try to tie them up in movement, as they lose the biggest advantage they have once it wears off in 5 seconds, Speed.
Snipers, very little overall you can do on the longer parts, but sometimes, you get lucky forcing the corner ones by launching a rocket or two to shift. If your closer, but not enough that you can hit them with a rocket in time, use the shotgun to foul the aim if they are on your level, the shells can give you the close range you need to use the rocket without taking a lead brain replacement. (Its not an always to note here, a good one can wait in that pause and get you, but it helps against most.)
Your biggest counter, are other soldiers, and duels are about prediction, and accuracy, the better of the two usually wins. (Where is he going, and where do i hit with rocket to send him up to juggle?) I can only guess to say, put yourself in his shoes, and ask, "What would I do in his shoes?" I hope that helps, but really, win or lose here, its always fun for dueling another soldier.
Now onto spies, and about detection. Its not all inclusive, and sadly, there are some truly great spies whom still find me a personal knife repository (Captain Calle, whom ive still yet to get a 1-1 kill/death ratio with) But the biggest ones i can offer, Is that no spy can use an unlock weapon in disguise. Spies as medics, do not have an uber gauge. Spies currently can see health when disguised, so get a bit more speculative when you are lower health, for many will pistol you for a quick kill rather than knifing. Watch ammo respawns and weapon drops from kills, if you see one mysteriously vanish, an enemy spy just nabbed it. Another spot, if your holding an area, always keep the ammo respawn out, to force a spy to wait or hold enough to push by or try to get you with low cloak. The spy in water, will make bubbles, out of it for a few seconds, will drop in cloak. (Easy kills both times usually) Keep an eye out for spies that hang on top of an engie dispenser to recharge cloak in busy spot (Jump through to test as habit, if your stopped, hit the dispenser with rocket) Another note, is when you see a friendly leap towards you from the stairs, jump back, as its a trick to stair stab for some spies as a quick kill.
Engineer, it depends first if you have the sentry guard, or the roamer. The first is often parked right behind the gun, and with dispenser supporting him health wise, both level 3 usually. Take out the dispenser first if possible, it robs him of a huge stash of health/ammo/metal. and usually can let you petter to the side of the gun to kill him, then the gun with ease. (set up he went with permitting of course) Calle can likely give you better tips as an engie how soldiers destroy guns, and the tips engies have to stop soldiers. I'm not really the best at explaining how i do it.
Next is dealing with attacking from the skies. Rockets deal a self damage of roughly 41 points, give or take a few points. Rocket jumping, and attacking from the air, you have the advantage hands down for two reasons, first is that your aiming angle is so much better for prediction (more on that in a moment) and that you are far far harder to hit on a good jump along with the fact not that many people look up. (If your above or have a hidden vantage point to fire down from) When you are looking down, the fact you can see how they are moving is much more like a map, and even where they want to go by weapon turning, or the backpedaling, and shoot so that they munch on mister rocket.
I hope these tips help, and enjoy a class valve wants to nerf more before too long.