Get 3 rune pages and fill them. One for jungler, one for top and one for support. As you pointed out, learn to play 3 champs per role. That is 9 champions in total. It is doable and this way you have a nice pool of champs to adapt to any situation.

Why I added support, well, when there is fill enabled you are probably going to be stuck with support sometimes. In this case you must be good as a support in order to be useful to your team and to avoid being a burden.

Practice how to cs without losing the minion gold (mele minion two turret shots and one auto, ranged minion one turret shot and two autos). Also be sure to get each and every minion, that is the golden rule, get as many as you can, each minion that you don't kill means less money in your pockets. Also, don't hit minions with full health when farming unless you want to push the lane. So many times I see people hitting minions randomly only to lose them to their own minions or turret shots.

Never use a skill unless it won't fulfill its goal. That means don't fire Ezreal's ult down the lane just because it is up, save it because you will need it. This is true for every skill in the game. Blindly casting spells just wastes mana and cooldowns.

When your teammates are doing poorly, instead of saying: "Stop feeding or x player feed, gg" you go to their lane and help them. By typing those hollow words you are not making the situation better, you are just making them perform worse. For those 40 to 60 minutes, your team is your family, help it and it will help you, if they are problematic, mute them to prevent further arguing and help them as much as you can.

One final thing, never, in any circumstance think about insulting or disrespecting your enemy. That will just create toxic environment. You will argue in the end and probably get reported. Be a gentleman, not an idiot.