Ah, the scout. One of the backbones of the competitive format, tied with the soldier for the most versatile class, capable of outmaneuvering any enemy and singlehandedly turning the tide of games. Mastering the scout can give your team a huge advantage, and reveal a lot about how to play the other classes.
The most important aspect of the scout is his movement. While positioning in general plays a huge role in TF2, as it does in its predecessor, Quake, the ability of the scout to double jump, coupled with his speed, give him an enormous advantage. Using the double jump, you can fake out almost any opponent, causing them to waste ammo and time, and give you a greater chance of killing them before they kill you. Obviously, given the scout's low health, it's vital that he avoid as much damage as possible, so place a high priority on avoiding damage, in most cases above getting kills. Not dying in TF2 is oftentimes more important than potentially getting a kill. Strafing one direction and then jumping in the opposite direction will oftentimes throw off people's aim. In addition, take advantage of the environment to the best of your ability. For example, jumping off and onto the train cars on the badlands mid makes you an almost untouchable target to everything but hitscan (bullet) weapons, because you'll constantly be in midair. The scout can use a height advantage almost as well as the soldier, so aim to get the upper hand (har har har) whenever possible.
But now for the fun part! The WEAPONS
The default scout loadout is one of the most effective weapon sets. The scattergun provides an excellent balance of damage and clip size, and the pistol is a fairly effective mid to long distance weapon. When using the scattergun, it is important to realize that while point-blank shots do the most damage, they also put you at great risk of taking large amounts of damage. Short- to mid-range damage output is still fairly high, and leaves you with more maneuvering room. Fans of the Quake series will recognize the similarities between the pistol and the LG, and will definitely understand the importance of good tracking. The bat, the default melee, has a faster swing rate than any other melee, but does half the damage.For a great example of how good scout can be in the right hands, check out this video, of probably the most well-known TF2 player out there:
As far as primary weapons go, you have two other options. The first, the Force a' Nature, is a shotgun-style weapon, with a huge damage output up close, as well as a knockback feature. Good aim is essential with the FaN, as is remembering that you will be stopped in midair if you should fire it while jumping. The FaN can also be used as a pseudo-triple jump, though it has limited effectiveness. One can still, for example, get on the roof of B on cp_gravelpit. One valuable practice technique is to use the FaN in 1v1 combat to improve your aim--because you'll only have two shots compared to the typical 6, it becomes vital to aim carefully and consistently.
The third scout primary, the Shortstop, is probably my most-hated weapon right now, besides the spy's Dead Ringer. It has a smaller clip size than the scattergun, and does slightly less damage, but has a higher rate of fire and has a slowdown effect. It is almost ubiquitously used by laggy players in my experience, and is well suited to a fast-moving, frequently jumping playstyle. Because of its relatively good accuracy compared to the scattergun, players with good tracking should definitely consider the Shortstop as a viable alternative to the scattergun.
The mad milk and Bonk! are the only two secondary weapons I will discuss, as crit-a-cola seems to be universally ineffective, in pubs and competitive play. Mad milk, with its jarate-style action, returns health to your teammates when they damage a "milked" player, in addition to extinguishing fires and revealing cloaked spies. It is excellent for rapidly clearing off a point that is crammed full of players, especially when coupled with the soldier's Batallion's Backup, as attacking players get both health regeneration and damage reduction. This secondary is much more of a support weapon, however, as the effects do not extend to the scout himself.
Bonk! is one of the most unique weapons in TF2. Drinking it gives the scout invulnerability for a limited time, with a slowdown effect afterwords. It can be used to distract sentries, confuse ubers, or physically block a choke. It is most effective when used in a coordinated manner, such as a 6v6 competitive environment, so that other players can take optimum advantage of its effects. As with the Mad Milk, Bonk! is primarily a support weapon; however, it can be used to escape otherwise certain death, so it is more than viable in any environment.
As far as melee weapons go, just use the bat or the fish. The candy cane and Boston Basher have downsides that are typically too steep to be overcome, and severely limit the scout. The sandman, once terribly feared and overpowered, now reduces the scout's effectiveness by making him a 1-hittable class, without any great advantages.
As far as maps go, the scout class is most useful in outdoor spaces, such as badlands or gravelpit, where there are several height levels that the scout can take advantage of. In tight places, or ones where the objective is extremely easy to spam, such as payload maps, the scout loses its effectiveness, as the possibility of instantaneous death is quite high. On CTF maps, the scout's effectiveness decreases over time, as each team sets up its defenses around the intellegence. However, by using Bonk!, a good team-playing scout can open up windows of opportunity for the rest of their team to break through the enemy's defenses.
Now get out there and play some scout!