My tips:
Going one for one is rarely ever worth it. Unless you are killing someone who has amassed a killstreak and therefore a large bounty on their head, or getting first blood, then it is rarely ever worth killing someone just to get a kill. Destroying a tower and dying is also generally not worth it as you can always keep the lane pushing later (this is called lane pressure, aka threatening to destroy an enemy tower by sending minions to it) and take it when the enemy isn't there to defend. Inhibitors and the nexus are different stories and are very big objectives to take most of the time.
Understand when you should and shouldn't chase. There's a difference between chasing enemies and getting baited and lured into a trap.
Last hitting minions and farming is the most reliable and usually safest method of acquiring gold and getting to lategame. Early game, don't try to depend on killing the enemy laner or damaging/destroying their tower. In the lategame, kills come by fast and towers explode quickly. Usually during midgame and lategame, people tend to abandon the minions in the side lane. These tend to build up into large "super waves" composed of multiple minion waves and are worth tons of gold and exp.
Wards are useful for map control. Map control means areas of the map where your team has vision of and therefore are locations that you can tell are safe or unsafe. They are also important for spotting jungle objectives such as Baron, Dragon, Red buff or blue buff. Try to ward an objective camp (if it is alive) as having knowledge of who took it is valuable information (or who is taking it at the time). Wards are also useful if an enemy runs into brush that you do not have vision of as they prevent brush juking. A well placed ward can also allow you to catch enemy setups and traps.
BE PATIENT, you don't have to be autoattacking something at all times, you don't have to be on the move at all times. Sometimes the best choice is simply to do nothing for a while and wait for opportunities to open up. This is very important in ganks.
Learn how to jungle. It does not matter what role you intend to main or play, knowing the jungle, the camps, how a jungler ganks, what junglers do, how to work with your jungler and why a jungler exists will help you out. Every role is affected by the jungler and every role benefits from knowledge of the jungle and what's important inside of it. The meta is shaped around the jungle.
Finally, understand gold checkpoints. This is something NO ONE EVER TALKS ABOUT. But what I mean by this is "how much gold do I need to finish my 
infinity Edge if, say I had a 
BF sword and a 
Pickaxe?". The answer is 1375 gold. THIS MEANS once you see that you have 1375 gold, you should recall back to shop and finish that item as it is a huge powerspike. Knowing the prices of items and how much gold you need can be the difference between actually doing damage and tanking during a teamfight or contributing nothing to the team.
My tricks:
There is a button to the left of the 1 key. It looks like "`" (or ~ if you press shift). In game this key is the "target champions only button. Use it to make abilities and autoattacks target champions only. Useful for attacking opponents in the presence of minions. You can also use it in the jungle to reposition yourself without stopping to auto attack the jungle camp, which is very useful for champions like Zac (picking up blobs) and Olaf (so you can put the jungle monster between you and the wall to throw axes that you immediately catch).
You can place wards over walls. If you think the enemy is taking baron or dragon, stand on the opposite side of the wall and put a ward into the pit. It's usually safe to do (since your opponent would have to jump over the wall to get you) and you get vision of the objective.
Right click items in the shop. This will immediately purchase the item. You don't have to double left click and maybe buy too many potions nor do you have to select and then click purchase.
does not affect champions, don't take it unless you want to kill big monsters.