Learn the champions.
For the most part you have to learn machanics. By this I mean learn how far your abilities reach, how fast their projectiles travel and when is best to use them, as well as the abilities of your enemies. When you know the speed of your stun/root/snare it is easier to aim it at where the enemy is going. Essentially making them walk into it. This also helps for when enemies are fleeing into the jungle and you lose sight of them.
Read the minimap.
Learning to watch the minimap helps alot. If you don't see an enemy on the map they may be moving to gank. Don't rely entirely upon your team for MIAs as even the best of players can forget at times to ping them. If you don't see the mid laner, there's a good chance he may be trying to sneak up on you. If you haven't seen the jungler in a couple minutes check the minimap. If you see him on bot lane you are good to continue farming and pressing lane. If you see him passing by mid then he's on his way to gank you. If you don't see him at all then pay attention to how your lane opponent is behaving. If they are staying towards the back of their minions instead of trying to poke you down, they may be trying to draw you into the jungler's attack.
Also when looking at the minimap gauge whether or not one of your team mates needs some help. If they do then try to help them if you feel you can leave lane.
Warding.
Take warding into your own hands whenever you have some extra gold. Warding is not just the support's job. use your trinket to place wards or clear them up, depending on which trinket you have, and whenever you have some extra gold go ahead and buy a ward to place somewhere. Make sure to ward gank areas early game, such as the river bushes, and the tribush, (the one shaped kinda like a Y). Mid game ward high traffic areas, such as the dragon pit, the Rift Harold, and the small bushes sitting in the middle of the river that you see Scuttles scuttling past. Late game start warding escape routes, such as inside both yours and the enemy jungles, and try to keep a ward at Baron and Dragon to keep a watchful eye on them.
Itemization.
Use items that compliment your champion's abilities, not just because you like their effects. 
works well on 
and 
but might not work so well with 
or 
. 
works well on crowd control tanks like 
and 
.
Also remember that one item set will not work in every match. You need to itemize according to the champions your are facing. If you are playing a tank and the enemy ADC is running a auto-attack critical strike build, (as most ADCs so.) or you might be facing a 
or 
building critical strike and attack speed, then building 
and will cut off 20% of their damage before you armor reduces it further. A 1000 damage crit would instead become 800, and if you have 71% damage reduction from armor it would be cut down further by 568 damage down to 232 damage. 232 damage damage sounds a lot better than 1000 when the enemy is also attacks at roughly two attacks a second.
Be open minded.
You can always learn something new, and learning each of the champions will only help you. Just because you hate a certain champion and want nothing more than for Riot to remove it from the game, doesn't mean you should close yourself off from playing that champion. Instead try them out. Learn how their abilities work and how they can be used. Learning how your most hated champion works will help you better fight against them in lane and team fight, as you can begin to predict how they will aim their abilities and when they will try to use them.
Ignore toxic players and don't be toxic.
If a player is being toxic just ignore him and focus and what you need to do and what you can do. If you find that they are distracting you with their comments and you are becoming irritated, then go ahead and mute them. It's much easier to play the game when you don't have to put up with people typing in all caps or swearing to their heart's content at you or a team mate.
Also remember not to be toxic to others. We all have bad games and you know it. You've had bad games as well and know what it's like, so don't take out your frustrations out on your team members. Getting angry only worsens your decision making and causes you to play worse. So why be angry? Okay so your ADC and support are feeding their butts off, mid keeps getting caught out by the enemy support roaming about, and your lane is getting camped by the enemy jungler. Instead of cussing and swearing and screaming at your team mates, instead just let all that anger and frustration fall away. It's much easier to turn the game around and cause the enemy team to throw the match if you keep a calm head.
However this isn't everything. Objective control is also a very important lesson to learn and target priorities. However I'll leave that to some of the others that might comment on this post to explain.
