While it really helps reading guides etc and learning what different champions do, it might be a bit daunting for someone new to the game to have to memorize so much information. There are however a few different things that can greatly help you improve with minimal effort.
1: Learn how to play defensively before you learn to play offensively.
If you're in doubt about your survivability, stick to your turret or at least far away from the enemy champions (exp-range is quite large). You won't get alot of last-hits this way, but you won't be feeding the opposing team either.
2: Throw away any crutches instantly.
Having summoner spells such as heal or revive just makes learning the game more difficult in the long run. However getting escape tools such as Ghost or Flash is highly beneficial (as those will actually remain useful throughout the game).
3: Every now and then you should read up on a new champion or two.
This is easily done at http://leagueoflegends.wikia.com/wiki/List_of_champions. Just reading about one each day takes merely a few minutes and improves your game-awareness incredibly. As for guides on champions you want to play more frequently, http://www.mobafire.com/league-of-legends/champions is a good source.
4: Practice, practice and more practice.
I recommend coop vs AI games until you hit around level 10. At that point you should start getting a feel for reading basic behavior patterns (example; enemy runs towards you = he probably intends to do something to you). It also helps you getting a good timing on basic abilities you should master, such as skillshots or defensive spells.
5: Have fun.
The whole game is for naught if you do not enjoy yourself. People often dis-recommend buying tier one runes for example, but part of the charm can be gradually beefing yourself up. It's the same reasons RPGs are fun. Do what it takes for yourself to enjoy the game as long you aren't ruining the experience for your teammates.
Lastly, I would also like to recommend Soraka for a starter champions. She is cheap and entirely self-sustained. Not to mention people will be loving you for keeping them alive with heals and giving out free-mana. Never underestimate the enjoyment one can have as a support character. his isn't WoW after all
Sorry about the wall of text