Basically, LoL has a massive, massive learning curve. When I came into the game, I came in with a very strong DotA background (mained support in open leagues with reasonable success, was generally pretty good at the game), so the fundamentals of LoL were fairly straightforward. I understood the UI in full because it's very similar to the DotA UI, and my background in RTS games had made learning DotA in the first place easier.
Anyways, something happened my sophomore year of college that made me think about both LoL and game design a lot differently. Basically, a buddy of mine was taking a digital games class with 4 other students, and for their project for the class, they all made level one accounts and started leveling to 30. However, while 4/5 people had 0 LoL experience, and I think 3 of them had relatively little gaming experience at all. my was gypsylord, who'd had his job lined up at Riot at the time (fun fact: I'm the guy gypsy referred to
in this post.)
Basically, he got a shitton of crazy insights about how new players approach LoL, because the game is crazy, crazy complicated. One of the girls in the group had only played like Harvest Moon, and the shit she would do when she was learning how to play was the sort of shit you would go "why the fuck would you even do that." Like, understanding how to use the minimap UI to go to places wasn't something she immediately got because she'd never played a game with a minimap before. The amount that she had to learn when she first started the game was incredibly ridiculous.
I was able to learn what every champ did nearly in full just leveling 1-30, because I started playing back when there were like 50 fewer champs. I got to gold S1 without having played most champs, and only being sort of unsure of what Morde did (I got the ghost and the aoe-dmg defensive buff thing, but I didn't get the difference b/w his two nukes). Nowadays, I don't think you can learn what champs do in full just by looking at them. Zed and Yasuo in particular are super complex in how they function, and Rengar's passive doesn't telegraph super-well as a non-ability to new players.
The first thing you need to learn in learning LoL really is the basics of the UI. Get comfortable with hotkeys, and move things around in a way that makes sense to you. Get used to not clicking on anything ever. Learn shit like using P for shop instead of double-clicking the shop. Buttons for items and B for back, not clicking on them (gypsylord used to click on pots until I yelled at him enough times to stop it). Learn how the minimap paths you so you can move the way you want to around the map. Learn what the map looks like, where the brushes are, etc. These are things that exist across all champs, and will either take not very long at all (if you're like me and had a DotA background), or may take you a shitton of time to learn (if you don't have a big video game background).
The next thing is to start learning is just what all the champs do, and this is a very ongoing thing. There are two ways to learn what something does: playing it, and playing against it. Obviously playing something lets you learn it faster, but playing against something will still show you what it does. Just take notes of what different champions do. What their abilities are like. As you see it more, you'll get used to it. This is a process; I'm still learning what the capabilities of a lot of champs are, and there are times when I'm surprised by how much damage something does, or how tanky something is. At the same time, I probably understand a few champs as well as, if not better than, some of the pros from how much I've played them. You can never really know what a champ is fully capable of, especially given how much that changes with balance patches, etc.
The next thing to focus on are the mechanical fundamentals. In order, these are csing, positioning, skill shot placement. This is the order in which you should focus on them.
CSing: Knowing how to last hit is fundamental, as that is your main source of gold on anything that's not a jungler and support. The way to practice this on it's own is to just make a custom game and try to not miss any minions while not using abilities. In real games, there are times when you want to use abilities to CS, and you'll learn that with time as well (and the decision-making aspect of that I'll get to in a bit). Different champs are easier/harder to CS with, so if you can CS with autos with Karthus or Soraka, you can probably do it with just about any champ. This is very, very important to get down well.
Positioning: Where you stand. This applies in the laning phase, as you want to position in such a way to protect yourself from ganks (other people coming to your lane to kill you) and their harass. This comes into play when you're roaming (leaving lane to go gank/take an objective) because it's how you don't get caught out. This applies in team fights because it either keeps you alive (if you're a carry/support) or determines how effective you are (if you're a tank/bruiser). This requires you to know what the enemy champs do, as a lot of positioning is determined by what the enemy team is capable of (i.e., if the enemy team has a Zac/Vi/J4, as an ADC I need to stand back more b/c they can all get on me from very absurd ranges and CC me).
Skill shots: Your ability to land skill shots. This involves knowing how to predict movements, and just gets better with time. Also, if you come from a fighting game background, you're really good at this by default (my diamond 1 buddy who plays competitive smash was basically a god at this).
After mechanics comes strategy, which is getting more advanced. It comes from watching better players play, and is probably not worth talking about until you get a handle on the other stuff.
