As an assassin, don't ever be in front. Don't think of initiating. Let the rest of your team do that, then strike. Doesn't matter who, as long as they die within a couple of seconds of diving on them.
As AP carry, be in about the middle of the team. When it starts, pour your combo onto whoever is A: squishy, and B: in range (preferably their AD carry, AP carry, or any assassins they have). Use any CC skills to help initiate if you can (
Gravity Field,
Dark Binding, etc.)
As Tank, be fairly aggressive, look for any mispositioning (see rest of these entries), and jam yourself in any crack you find as long as your team can follow up.
As AD carry, sit just behind the AD carry and machine gun at whoever is in range. If your only valid target is the tank, shoot him, but prefer other targets.
As a tanky support, get in there and soak some damage while saving allies.
As squishy support, stick on the edge of the fight and use your skills as needed to help your team.
There are exceptions, like Ashe should be looking for moments to start fights with her arrow, Morgana is often best used as flash-ult-zhonya's while your team corrals everyone around her, things like that. But that's a pretty good general guide.
As AP carry, be in about the middle of the team. When it starts, pour your combo onto whoever is A: squishy, and B: in range (preferably their AD carry, AP carry, or any assassins they have). Use any CC skills to help initiate if you can (


As Tank, be fairly aggressive, look for any mispositioning (see rest of these entries), and jam yourself in any crack you find as long as your team can follow up.
As AD carry, sit just behind the AD carry and machine gun at whoever is in range. If your only valid target is the tank, shoot him, but prefer other targets.
As a tanky support, get in there and soak some damage while saving allies.
As squishy support, stick on the edge of the fight and use your skills as needed to help your team.
There are exceptions, like Ashe should be looking for moments to start fights with her arrow, Morgana is often best used as flash-ult-zhonya's while your team corrals everyone around her, things like that. But that's a pretty good general guide.
From my expierence, the 1 line tips for each role:
Bruiser/Solo Top/Junglers: Sit on their carry (the one with higher damage output) to your death, drive him out of the battle.
AD Carry: Get their squishest targets down at once, other AD Carry > AP Carry > Support.
Supporter/Jungler/Disrupter (tons of CC): Stay on your's Carries backs and defend them at all costs.
AP Carry: Throw out your nuke combo at everyone and focus on killing AD/AP Carry.
Bruiser/Solo Top/Junglers: Sit on their carry (the one with higher damage output) to your death, drive him out of the battle.
AD Carry: Get their squishest targets down at once, other AD Carry > AP Carry > Support.
Supporter/Jungler/Disrupter (tons of CC): Stay on your's Carries backs and defend them at all costs.
AP Carry: Throw out your nuke combo at everyone and focus on killing AD/AP Carry.
^ That's right except for the AD carry part. Good luck hitting the opposing squishies without dying.
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All too often I find myself dying either cause I engage without help or being too late into fights or simply don't do the right thing.
Basic knowledge of Lane is easy: last hitting, ward, choose when to push and when to farm passively, help other lanes.
What are the same tenements for teamfights? What does I need to know theoretically to improve and actually learn from my experience?