First play the tutorial and bot games to get a feel with the controls and mechanics of the game.
The tutorial is really quite good, and should be played by every new player.
Learn to use the hotkeys q w e r for spells, d f for summoner abilities.
Don't take revive.
Seriously don't.
It has it's place in twisted treeline RARELY, but even then take abilities that will keep you alive or benefit the team
Learn to deliver killing blows to creeps or "last hit"
Play defensive.
If you are doing damage to an enemy near their tower the tower WILL attack you
This applies with damage over time effects like teemo's poison or ignite.
Practice with a character before going into actual matches.
Again, don't take revive.
Assign your mastery points you have one point at level one use it.
Buy a solid set of level 1 runes and then wait until level 20 to upgrade them into level 3 runes.
Press b to teleport directly back to base, however if you get damaged during the time it takes to blue pill (return) the teleport will be canceled and you have to try again (so don't do this out in the open [normally])
The Teleport summoner skill is a great, it allows you to heal/regain mana and get back to a lane, stop an enemy push across the map, get to a team fight quickly.
Again, don't take revive.
Exhausting a physical character like ashe, tristana, or master yi will make them near useless in a team fight for 3 seconds due to the blind (3.5 if you get the mastery)
If you take exhaust get the mastery for it. It only takes 1 mastery point to get it.
Ghost is a great skill for getting away, chasing, or getting somewhere fast.
Defeating the dragon on summoner's rift as often is possible greatly benefits the team. It gives 130 gold and experience to each player on the side that killed it everytime it's defeated.
Pay attention to the minimap it shows allies, and visible enemies in their locations.
If enemies are missing be careful, tehy may be coming to kill you.
Wards are a great investment, because they allow you to see an area of the map for a certain amount of time.
Sight wards see an area of the map, Vision wards see invisible units too but they are more expensive.
Again, don't take revive.