I play scout competitively. I'll point some things out
Initally, try focusing on having really good aim and hitting every single shot you fire. If you have trouble aiming with scout i'd advise turning off mouse acceleration and checking raw input from the mouse settings in game. You can also lower your sensitivity as it will give you a wider range of margin for error when aiming. I personally use 9" for a 360 turn in game but you may be comfortable with 1"/360 - 4"/360. Make sure you turn on "play a hit sound every-time a injure an enemy" and "display damage done over the enemies head" from advanced options.
A close range scattergun shot does 104 damage, while far away it does 3 damage. Pistol is 22 damage per shot up close and 8 damage far away.
When engaging scouts with good aim it is dangerous to jump and most scout fights are pretty equal. You should install the null-cancelling movement script in the guide for scouts above and press A + D to strafe to the left and right to dodge the other scout while trying to hit him.
When engaging soldiers try to use the environment of the map to your advantage, try to keep high ground by using crates/rocks/stairs etc. Pester him with mid range shots and go in for a "meatshot" (100dmg) if it is safe. Try and make him waste his 4 rockets on you before going too close. Use your double jump on and off of crates, if the solider hits you, you can strafe in the air, land of something and continue engaging him. Engaging demomen is similar but easier you might not even have to use high ground depending on the skill of the demoman.
When engaging heavies you want to catch them by surprise, if they are spun up and looking at you then they will probably kill you. Try to use a flank and get close to the heavy without him seeing you and deal 3 or more meashots. If he turns around try rounding around him while shooting or using scout vs scout tactics with wider strafing angles.
When engaging pyros you have two options. One to run backwards and to hit him with mid range shots, second go in next to the pyro to deal 2 meatshots then get a health pack. If a pyro airblasts you into the air, you can use your double jump to have advanced airbourne movement and can engage him similarly to a solider.
Other classes are quite easy to deal with, you just have to remember how much health they have, how much health you have and how much damage you can do to them.