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You didn't go to Med School, did you?
The medic is the backbone of a TF2 team. A team without a medic will lose every time to one without one. Conversely, a team with too many medics will have trouble advancing, and will likely create a stalemate situation. Playing a medic seems simple at first, as you don't have to worry about aiming at enemies, but it is in fact one of the more complex, challenging and cerebral classes to play. That is, unless you're ramming your bonesaw into someone's ear canal.
The medic achievements aren't easy to get (unless you're playing on an achievement server). They require you to remain alert and attentive to your own health while keeping your teammates alive for extended periods of time. It requires that you know exactly when to ubercharge and how to pick the right person to team up with. The best medics are able to heal multiple people at once, switching their heal beam to catch the people who are in trouble without skipping a beat. You shouldn't be stuck healing the same person throughout the whole match. It's a waste of your ability and limits your team's effectiveness. The more healing you dispense, the less that your team will die and the further that they will be able to push up in shorter amounts of time.
You should be using your medigun constantly, except when you're alone and need to defend yourself. Some of the best targets for healing are heavies, soldiers and pyros depending on the map you're on and the task at hand. Soldiers are especially good at destroying sentries and obliterating tightly packed groups of people. Pyros are excellent at clearing out close quarters. Heavies are good for taking out pyros and generally gunning down people from short to medium range. Ubercharges should be reserved for when the person you're healing is ready to move forward and knows exactly what they're going to do. There's nothing more deflating than a wasted ubercharge - use them wisely!
One of the most important skills that you need to learn as a medic is when to retreat. When you have the option, let the person you're healing die before you do. You need to save up your ubercharge - their life is worth less. A good dynamic between a medic and the healing target is one in which they both pay attention to each other's health, retreating back when they need to heal up some more. This makes it easier to go for a big push, as more characters will be alive and at the front simultaneously than if you were to just continually advance mindlessly.