Best way to improve in LOL?

What is the quickest most effective way to improve my lol play?

Training, mainly.

But how, you may ask? Through some other games as well. Osu is a good example on how to improve reflexes.

Watch your replays, thats how you improve the most.

I found that playing 2 games to warm up before ranked and experimenting a little with playstyle has helped me get out of silver.

What I have found insanely helpful, and what lead me climbing from silver elo to high plat and even diamond is watching coaching streams, you should look up LS on youtube, this guy has really great videos, where he talks about simply the fundamentals of the game and their importance, then going more in depth when coaching Higher Elo players concerning micro and macro play. And, honestly, a really cliche thing to say, but just play more, there isn't a single competitive game where you are the best when you start to play, you gradually improve, learning different concepts of the game as you play.

I'd say the best thing to do is to focus on micro rather than macro concerns. Trying to get a little bit better at everything at once is nice, but it makes it harder to identify improvement.
If you play adc, focus on last-hitting for a while, either in bot-stomps or real games. Try to sort out how much damage you actually do to different enemies at different levels and with different items.
If you play something with a lot of skill-shots, try to focus on leading your shots. Try picking up whatever the fast-casting thing is.
If you play jungle, sort out good ward positions and try to work on clear times.
Break your gameplay down into components and focus one at a time, and even if it takes a while to meld them together, you will see improvement in areas as time goes on.

I would like to recommend pro players + couches. Also it's great option to read guides with wave managment. it's pretty easy and you will improve a lot!

Every time you finish a game open your notebook and write all the points that you could improve and all the mistakes you did and how to improve them

practicing and playing the game for a while is the key i guess

This video helped me out a lot! It's very in-depth and practical. It's on YouTube titled "If you ACTUALLY want to get better at ADC, watch this 1 hour video".