This is a compilation of things I've learned playing, watching streams, and bouncing around in the forums. Starts off easy and gets more specific. Not necessarily in order of importance.
I posted this in General Discussion and got some feedback, but I can continue adding to this upon request. Ask me anything!
TLDR: Go through and read the bold I guess.
----- 1) Have good team composition. This happens before the game, and is extremely important. An ideal team has at least one tank, one AD carry, one AP carry, and a support. A jungler is basically required. The US metagame ("metagame", simply put, is the most effective playstyle most people adopt) usually has carries laning solo. Don't have a main. Learn to play many, hopefully at some point all, champions. You can exploit their weaknesses, and use the same methods that were used against you. Learn to play carries, tanks, and support. Learn how to lane well and how to jungle well.
1.a) A tank is not a sack of meat. Real tanks have CC and initiation, but those two qualities really serve only one simple purpose: to protect their allied carries as they drop enemies. Champs like Mundo and Mordekaiser are better described as "soakers" and "off-tanks". Please build some damage on champs like these or else the enemy team will simply ignore you and kill you last since you have almost no impact in team fights.
1.b) A carry scales extremely well with items but should always build some survivability. A glass cannon is great for soloing baron in a practice game, but in ranked team fights will be evaporated by a single ranged ability. Banshee's Veil recommended.
----- 2) Last hit. (Early game) One of the first basic concepts to understand in laning, last hitting is essential in not getting ganked. Constantly auto-attacking pushes your lane towards the enemy tower, and leaves you either overextended or underleveled. Learn how much damage your auto-attacks do and practice hitting at the last possible second to get those sacks of gold. Hit tab or whatever you have it bound to to compare CS.
2.a) Gank. (Mid game) The jungler is not the only one who can gank. If the champion you are playing is a better ganker and your hands are free, if you are mid solo and the enemy champ went b, if for any reason you see a good opportunity to gank at this point in the game, do it. Getting a few kills up early/mid game gives a huge advantage late game.
2.b) Then start pushing like your life depends on it. (Late game) Your life does depend on it, though, so don't be pushing by yourself when 4 enemy champs are mia. Point being, in the end, this is not a game to see who can get the most kills. Whoever destroys the other team's nexus wins, simple as that. Create diversions or have 4 teammates *cautiously* push mid while the great pusher (Sivir, Yi, the trio of global tele's: TF, Panth, Shen, etc.) off-pushes. There are different ways of pushing, but pushing is what wins games. Another important way to think about pushing is to force them to react to your movements and pushes, instead of the other way around. When a team can only react to the other team, they are forced to play more defensively, which gives the aggressive team a slight exp advantage and a HUGE map control advantage.
----- 3) A very basic concept that is very difficult to master: map awareness. This entails many, many things, but there are several important ones. Calling mia on enemy champions early game, even if they just went back to their fountain. Placing wards. Having a clairvoyance is great. The US metagame has very late game calling for the travel in relatively close groups of 5. Branch off to stop lanes from autopushing, but stay together or be able to get together quickly.
3.a) This should probably be a whole topic but whatever. Ward. Everyone ward. In early game it is the responsibility of the jungler to ward dragon. Laners and junglers share responsibility to lane near the river. However, as soon as lanes break down and midgame begins, it is everyone's responsibility to buy some wards.
3.b) Laners really need to only worry about their rival laning champs and the enemy jungler in early game. Junglers, on the other hand, should be aware of the status of every lane. If an allied solo champ dies or has to b, the jungler should take care of the lane until they return. The jungler should know when an allied champion needs or should get either red or blue buffs, and cook (start killing) the minion camp so it doesn't take long for the carry to get the buff and return.
----- 4) Positioning and team fights. One of the final and most difficult concepts to grasp and master. There are several basic premises. Tanks should always place themselves in between the central fight and their carries. Carries should put themselves at a safe distance from CC but should attempt to take out higher value targets, i.e. AP/AD carries. Included in this topic is initiation of team fights, the concept of focus, AOE, and general positioning in scouting and in team fights.
4.a) After positioning, focus is the next important aspect of team fights. Deciding on priority targets and killing them first as a large group or as smaller groups of 2 or 3 will win you team fights, end of story. A team that has their 5 champs spreading their damage and abilities randomly to their nearest enemy will hurt everyone but kill no one. A 5v1 attack quickly ends (do not blow all your abilities just to kill one champion) and the team fight is 5v4, insanely in your favor. Probabilities for enemy champs to start running probably triple after a teammate is slain, and likely in scattered directions, making them easier to pick off.
4.b) Initiating is a huge factor in winning team fights as well. A good initiation can straight up win a team fight. Malphite flying in and knocking up 4 champs is a great start. Rammus powerballing in and taunting their Ashe is a great start. Do not initiate if your team is not ready. Ping before you go, and make sure allies look like they are gearing up. Make sure it's not 4v5 them. Go down the line on the left side and check for green lights, signifying everyone's ult is ready. This is the biggest chance a tank has to shine, so don't blow it.
4.c) AOE (area of effect) has always been a great addition to the team. Whether it is an aura or an ult, AOE has been one of the most consistently strong metagame components throughout LOL history. Champs like Amumu, Fiddle, and Galio (just to name a few) obviously need to get as many enemy champs in their AOE ult as possible. In 5v5 team fights a lot of AOE causes utter chaos, so learning to identify your position in fights like those simply takes time and experience.
4.d) Positioning is probably hardest for ranged AD/AP carries, since they are probably the squishiest but are most vital in winning a team fight. There are many tricks, not the least of which is flashing over a wall to start sniping/assassinating from relative safety or getting an item like Zhonya's Hourglass.
----- 5) Common courtesy. It's fine to not say "gg" at the end of every match if you didn't think it was a good game. Don't think that the winning team is gloating if they say "gg" and don't be annoyed when a losing team says "bg". You've probably been in everyone's position at some point so try and have some perspective. Don't rage quit, afk, or incessantly troll. Explaining how a player could improve is better than saying he/she sucks. Obvious things.
----- 6) Miscellaneous:
6.a) Give your carries the kill if you can. They scale better with items so they need the gold more. Simple.
6.b) Several tips for team with jungle (better be all teams):
Guard your jungle! Top and mid (or bot and mid, depending on whether you are on blue or purple team, respectively) have nothing to do while they wait for creeps to spawn, so there is no excuse. One champ from the 2 lane could also come. Additionally, if they don't have a jungler (or a weak one) it might be a good idea to start by invading them. If your jungler is starting at blue, it's always a good idea for mid to hit the golem once, just to draw aggro. Mid champ can immediately run away, and the golem will wander a bit before starting to attack the jungler.
6.c) Jungle respawn times:
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