Read those two threads. They may be ancient but it still applies now.
League of Legends is a lobby based game. Once you are matched into a game, you enter the "Champion Select". Here you choose a champion, or character, to use for that match. Choose summoner spells, which are located next to your masteries and runes box. You start off with a limited choice of spells. After everyone has chosen and locked in or the alloted time has passed, the loading screen for the game will start. This may take awhile depending on the slowest person's connection. I remember it felt like forever my first game :)
There are 3 lanes (top, middle, bottom). Generally, one champion goes mid, two top and bot. If there is a jungler (champion who kills neutral creeps instead of laning), top will be solo. I will go into the benefits of this later.
You start with 450 gold, which can be used at the shop next to where you spawned. Where you spawned is the fountain and being in it gives you extremely high HP and MP regen. You'll want to use most of your gold on starting item. Look up your champion at leaguecraft.com or mobafire.com (preferably during the loading screen) and find a well rated guide.
You have 4 skills, 3 normal skills (Q,W,E) and 1 ultimate (R). You can add a point in your ultimate at levels 6, 11, and 16. QWE have a max level of 5 and your ult has a max level of 3. Add your first skill by clicking on the button above the skill.
The map is divided diagonally by a river. There are two "turrets" along each lane for each side. They do significant damage early on. Then there are the team bases on the top right and bottom left of the map. Each lane has an inhibitor and an inhibitor turret near the entrance to the bases. Then there is the nexus and the nexus turrets. Killing the nexus wins you the game.
Minions spawn in waves in each lane. If the inhibitor is destroyed for a lane, a super minion will spawn each wave. Inhibitors respawn in 5 minutes. You'll want to protect them :P
You only gain gold from landing the killing blow on minions so you need to practice last hitting. Last hitting doesn't push your lane so you are safer and the enemies need to come closer to your side which leaves them more vulnerable. After the laning phase, champions or entire teams will start to roam and cause teamfights. See that red cross on your minimap? That's Baron Nashor. He's a neutral creep that when killed, gives the killing team an amazing buff. There are other neutral creeps that give a buff, but only to the one champion who lands the last hit. They are the Ancient Golem (commonly referred to as blue buff, blue) and Lizard Elder (liz, red, lizard, red buff). The Ancient Golem is located at the upper left of the blue team's jungle and the bottom right of the purple team's jungle. Jungle is just the forest inbetween the lanes and they contain creep camps. Lizard is at the middle camp of the bottom side of blue team and the upper side of purple team.
For your first dozen games you'll probably be really confused. And you'll get a lot of hate. But if you can make it past the shit community, you'll find a great game. :)
Note: This guide is mainly geared to the 5v5 Summoner's Rift map, though some things still apply to 3v3.