Know if your champion has an AA reset, and use it effectively.
If chasing an enemy through a bush, ward the bush as soon as you're in range. If you don't have a ward, hit A to attack-move into the bush. This ensures that you will move directly towards and/or AA the target as soon as you have vision on them again, even if they juked you while you couldn't see them.
Carefully watch which enemy minion is being targeted by your siege minion. That target will take larger chunks of damage and will die faster. Try to notice which target it switches to each time its current target dies.
If a wave of minions is incoming to your tower, immediately assess which ones will not be killable in 1 AA between tower shots. For example, a melee minion at 75% health or a caster minion at 50% health. Plan out which minions need to be "prepped" with some damage and in what order, or which minions you should use abilities to kill. Towers will target siege minions before smaller guys but they will not immediately switch targets as soon as a siege minion comes in range.
If someone on your team has long duration CC (Warwick, Morgana, Rammus, etc.), do not immediately use your CC on the target that they catch. If you are Pantheon, for example, don't use your W to jump on an enemy who just got caught out by a teammate.
Ping OMW when heading to a lane to gank it, then ping 2 or 3 times right when your want your teammate to engage. If he wants to wait longer to engage, he'll disregard your pings. But if you don't ping at all, he won't necessarily notice that you've arrived in the lane.
Stay positive! 