Having grenades isn't unique to the Medic class. However, the nailgun and the bioweapon aren't enough to take out proficient members of heavily armored classes. The key to grenades is to prime them so that they blow up right next to a foe. When following an enemy, throw the grenade a little earlier so that it has a little bit of "travel time" and blows up near the enemy.
Once you press the +gren1 key, the grenade WILL detonate in 3 seconds. You don't have to let the grenade go as soon as you press the button. Instead, count to three, then throw the grenade. (When throwing concussion grenades, maybe a count to 2 and a half is best). When a scout is running toward you from the bridge, begin priming your grenade. Shoot some nails to slow him down. After around two and one-half seconds, throw the grenade in his general direction. You'll be surprised at how many times this works.
Grenades play an important role in jumping. You may not know that the medic can concussion jump. This tactic involves throwing a concussion grenade; the jumper should be in the air slightly in front of the explosion. With practice, you can jump very high.