First thing to learn when beginning League of Legends
Map Awareness
Riot has garnered some pretty effective videos with their Video Tutorial contest (Links posted below). As a beginner, you want to primarily focus on making sure you don't get ambushed by the other team (called a "gank"). While some of this involves Map Awareness, you also need to learn to not overextend, as well as keeping an eye on the enemies you're fighting in your lane. If you cannot see one or both of the enemies in your lane, let your teammates know!
Last Hitting
Killing minions and enemy players nets you gold to buy items. However, if you don't land the killing blow on an enemy minion, you will get ZERO gold, even if you've done most of the damage to it.
The faster you kill enemy minions, the faster the minions reach the enemy tower, which will make quick work of all non-cannon minions. It's better to keep the minion fights as close to the middle as you can; you only need to finish off the minion to get gold from it -- see if you can limit yourself to only attacking minions when you will kill them!
Tower Aggro
The enemy towers are dangerous. They have lots of health, armor, and a powerful attack that can quickly decimate low-level heroes. It's important, then, to know how the turret picks its target.
Enemy towers will always target allied minions first, switching to champions only if there are no longer any allied minions around. The exception to this is if you deal damage to an enemy champion, in which case the tower will immediately switch to the damaging champion, shooting at them until they die or move out of range.
Let me repeat that: If you deal damage to an enemy champion within range of the tower, it will immediately start shooting at you: You do not want this. Be very careful when attacking the enemy at their own tower.
Recommended Viewing
Four of these videos are a result of the Valoran Video Contest held by Riot Games, and are quick, 1-minute videos touching on just the basics (which seems exactly what you're looking for). The fifth video is made by Shurelia, an employee of Riot Games, and goes much more in-depth into what "Zoning" is, and how to take advantage of it.
Don't Die (aka Don't Feed)
First and most important rule in League of Legends is to not die, or at least, not die easily out of a team fight.
Your death cause a lot of troubles to you and your team. While you are dead and waiting to respawn:
- You don't acquire experience for minions kills.
- You don't acquire gold from last hitting minions.
- You leave your lane and risk to lose the tower (early-mid game)
- Your team is disadvantaged in team fight due to inferior number (mid-later game)
In order to avoid dying easily remember that your towers are nice friends especially in early game; move near them, while recovering health/mana.
If you spare some gold buy a couple of wards in the first teleport back to your base and put them in the bushes to avoid being easily ganked (attacked by an enemy coming from mid or another lane and basically attacking you from where you dont expect them to attack you from).
In addition, I just found this guide that I think is good to read for a beginner: 10 things every League of Legends player should known.
Positioning.
Actually, this is basically true for just about any team-based game - if you're in the right place, you're helpful regardless of how unskilled you are as a player. If you're in the wrong place, you're either contributing nothing, or getting butchered by the enemy team, and this is true regardless of your individual skill level.
With regards to LoL specifically, my suggestion would be to find a good, general purpose build, then print it out and tape it to your monitor. Follow that build exactly, every game. It won't be optimal, but it will be "good enough" and let you basically ignore that part of the game and focus on learning the important stuff. Like positioning. Then once you're comfortable with that, you can start to investigate different builds.