It's a fairly simple game to grasp, but it'll take a while to master. When you're starting out you just need to focus on smaller things such as last hitting enemy minions properly since you only get gold for being the one to land the finishing strike, learning your champion + skills to level up, what the items you can buy are, what and where buffs are, and not dying/watching the minimap to see where your enemies are (basically the same thing).
From there, you just need to know the general concepts behind each champion, like what their normal role is on a team, what Crowd Control they have (stun, blind, knockup, slience) [AND PASSIVE ABILITIES, when I first started out and used Anivia, it felt like none of the other new players knew about the whole rebirth thing] and you move from building your champion to being able to counter-build and counter-pick champions to give you an advantage over enemies.
From there, it's just timing, working with your team and learning more champions.