a helpful tip i have is that when you see a champion you want to try, you should first try him/her in coop vs ai first to get a feel for it. pull up a guide to get some tips on runes and masteries. most of the top guides also have detailed sections as to what your role in the team is.
another vital part is knowing your role in the team (^):
-if you are the support, harass the enemy laners and dont take the minion kills from your ad carry.
-if you are the ad carry, focus on last hitting and occasionally harassing.
-as a jungler, clear your jungle and gank.
-as top, focus on winning your lane, as one kill makes the difference up top.
-mid...just farm and get fed.
the MOST IMPORTANT THING in this game is to NOT PLAY THE BLAME GAME. (unless your blitzcrank pulls an amumu into your team. then scream at him forever) the more you play the blame game, the less you are acknowledging your own weaknesses and flaws, and therefore you wont improve. if other people blame you for everything, just feel good about yourself when you get better than them :P