As Poppy is my favorite champion in this guide i will review how to properly skill, rune and play poppy on any skill-level. I will base my facts and opinions accordingly so you will get a clear overview on how to play AD & AP Poppy.

Poppy is a champion who is highly underrated but can be devastating in the hands of the right person. She currently doesn't work well in the standard EU meta with top bruiser-ap mid-support/ad carry bot, as she will be completely dominated by any decent bruiser (Jarvan, Irelia, Warwick, Udyr, the list is long), however the US meta still works, as poppy is an extremely strong champion with an incredible burst and very good roaming abilities.

Pros:
Extremely high burst
Chase & Run extremely efficient because of her W
Can turn a 5v5 into a 5v4 in a matter of seconds
Snowballs very easily if fed

Cons:
Extremely farm/kill dependant in the early phases of the game
Can be poked/ganked very easily pre-6
Requires a support in bot lane to be efficient

Poppy's main role in team fights, is to wait. You wait untill you got an opening to either come from the back or the side, pop your ult on a carry and burst them down thus making your team have a great advantage. In early/mid game you want to go for the AP carry, since he/she is most likely the one with the most damage at that point of the game, and with the standard triple dorans into deathcap build, you should have no problem bursting them down. In late game, you want to focus the AD carry, in order to stop the enemy team from pushing.

Typically skill order in team fights is:

W -> Run into range so you can pop R -> R -> E -> Use Deathfire Grasp -> Q. If the carry used CC on you pop your quicksilver sash and do your thing.

Boom, 1 person is dead. Now if you're good you can get away just by running, else you wanna use either ghost or flash to catch up with your team, if you haven't initiated fully yet.

Poppy isn't a very good initiator as AP because you will normally only be able to burst down 1 person before your ult runs out (you pop your ult on the support and kill a carry, ult will most likely run out before you got your burst CD's ready again), so just kill 1 person and get the hell out of there.