My suggestion would be first to figure out what type of characters you like to play. me personally, i like playing characters that have a decent AOE, like Karma, Morgana, and Nasus. I also like playing a few random characters like Nautilis, Ashe, and Riven. Find out what works best for you when it comes to buying a character. The free rotation characters a re a really good place to start, but after a while your going to realize that there are some characters that don't rotate nearly as much as others.
Once you have an idea on what your looking for, do a little research. Personally i like to watch the Champion Spotlight on them to get an idea of things like moves, builds, strategies, etc. You can also use mobafire.com when you've gotten a character that you want to expand with to learn different builds and get a better idea on what works best for that character.
As far as getting better, fortunately from what I've learned and seen, the biggest curve comes from the characters. Learn the characters that you play, then learn what characters counter yours. This will give you the biggest advantage when and if someone tries to counter pick you in a game.
Next is to find people that compliment you and your play style as well as you complimenting theirs. This is a team game, and if there is even one person that doesn't help the team then it makes things much harder. Find players that like to have a good time. understand that there are bad games, and that communicate. Communication is key in a game like this. Learn how to call Barron, how many enemy's in lane, who they are, when, who, and when to take buffs. When to push, when to cover, ward, gank, support etc, etc. The list continues. Only then will you really start to see how great this game can really be.
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Finally, Learn to let go. Ive played plenty of matches, like today. I know how to play Nautilis in such a way that it compliments my personal play style. But for some reason, i was having a terrible game. No matter how much you practice, there will still be games when it just seems like nothing comes together for you the way you want it to. It happens. Just remember that it is only a game, and the point of playing a game is to have fun. Be kind, be friendly, and have fun.
I hope this helps. And please, by no means think that i am a pro of some kind because im not. But i do understand the value of team work, and ive played with enough great players out there to learn exactly what your trying to learn. Ive played local tournaments in my area and ive played in other eSoprts games online and the lessons are always the same. I hope that this helped at least a little, and if you have any other questions please feel free to ask and ill do everything i can to answer to the best of my ability.
xxTakeda01xx