If you are playing support and want to climb, work on looking at the mini map, learning to play the macro side of the game, and understanding cooldowns.
There are two ways to climb in this game: Mechanics and Game Knowledge.
Mechanics is the little stuff your champion does. Attack moving to kite, flashing a Malph ult, comboing properly on your champ, using Heal/Exhaust/Ignite/etc at the right moment, etc. Those are all mechanic things you can do that will impact the game, and most of them are stuff you have to keep in mind in case it happens or react once you see it.
Game Knowledge is the bigger picture. The whole point of the game is to kill the enemy Nexus. It doesn't matter if your team has 0 kills to their 500, if you kill their Nexus, you win. Knowing stuff like that buffs respawn around 5 minutes into a game, that Rift Herald despawns at 19:45 unless engaged in combat, or how to play when important cooldowns are down.
You can climb by having either of these, but both is obviously going to make climbing easier. Game knowledge is something that supports need to have though, cause they will be the ones looking at the map most of the time and can make plays based on that. A support is rarely going to solo kill an enemy. Obviously you can outplay, but it is generally unlikely. However, any support can set up a slow pushing minion wave which can take a tower if no one stops it, which opens you for your team to get other things like baron or dragon while the enemy is dealing with that wave.
Using game knowledge well is a less risky way to win. You could misplay a teamfight, but you really can't misplay having minions take a tower while you take another tower and forcing your enemy to decide which to defend.
Does the enemy have basic ability CC (Blitzcrank, Morgana, Jinx, Veigar, Ahri, etc)? Do they have multiple of those people?
Look out for an invade then. Position so that you can see them coming without locking yourself into a bad position. On red side, stand in the plus near blue buff closer to your tower, NOT in the bush by blue buff. That way you can see them coming from both angles and give yourself two ways of escaping.
Did the bot lane leash?
If you help your jungler and the enemy bot lane arrives to lane around the same time as you do, then most likely their jungler started bot side buff. That means your top laner should be careful around 3 minutes into the game cause the jungler may go top lane and gank them, especially if they are pushing.
If the enemy bot lane does not leash though, then you should ward around 3 minutes just in case the jungler tries to gank you.
Who is going to get level 2 first in bot lane?
You need the experience from the first wave and the 3 melee minions of the second wave for level 2. Bard will always get level 2 before that because he gets EXP from picking up chimes. If you pay attention to the minions, you'll know who will get level 2 first and you can either engage when you get it, or back off if they get it.
Those are some very simple things you can think about each game, and those are things for only the first few minutes of the game. There are tons of things like that to consider that don't require mechanical skill that can help raise your chances of winning.