Another tip: if you can, sneak behind and wait until the enemy is running away from you in a straight line. Line up your reticule, and then two clicks with the scattergun will usually result in a kill.
IMO the hardest part about playing scout is moving at top speed/strafing WHILE still being able to keep your reticule on a moving target. I do the opposite of what I do as sniper: I drag my reticule and once the enemy passes through it, I click. DO NOT PANIC! Every time I don't get the easy kill on the medic or lose to a scout (not necessarily 90% luck imo) I know it's because I probably missed the first 2 shots, then panicked and started unloading all of my shots in the hope that a stray pellet would hit and I'd get the kill, instead of waiting until my reticule was on the enemy.
The scattergun is really, really powerful but ONLY if your reticule is pointed DIRECTLY at the enemy. It's like using a shotgun except you move so much faster and reload so much quicker it's a lot harder to aim and/or actually take the time to aim before you decide to fire.
The pistol? Use it once they get out of range and are still running in a straight line away from you! Usually when they are making a beeline for a doorway. Strafe until they are directly in front of you and unload.
The bat? Use it on snipers so you can get yourself served. And/or soldiers like luuser says. It does wayy too little damage to be effective unless you know the soldier is at low health and panicking to reload. But if the soldier is smart enough to pull out their shovel...
I agree with santa that scouts aren't suited for a lot of maps. I LOVE playing scout on stag, fastlane and turbine where you have wide open areas for the center and multiple flanking routes for the rest of the points (well, not so true on turbine...) You win when you are able to circle wide and get away because of the large open area.
And yes scout is a great pyro counter in these wide open areas. Cause usually pyros stare straight ahead and run in a straight line when they're focused on someone else besides you
Santa's a big fan of always jumping and changing direction in midair; if you're skilled enough to focus on doing that AND targeting accurately at the same time, it's great. Unfortunately, my brain can't seem to do both tasks at the same time. Ideally yes, changing direction is one of your greatest assets as a scout because it's a great way to dodge rockets and the like. However, although it increases your chances of survival, it makes targeting accurately a whole lot harder, so you end up cutting your offensive power.
My suggestion is to focus on aiming accurately first, and then once you get it down, try doing some random jumps here and there and see if you can still track people well. To be honest, I think sniping helped my scout skills because of the emphasis on aiming. And playing scout helped my shotgun skills
Usually people pull out the shotgun as a panic weapon and shoot randomly hoping that again, a stray pellet will hit. However it pays to calm down, wait and aim accurately.