So... let see

You have the LoL client installed?

right?

ok, you download the game you have a shortcut

"play league of legends"

then if you exe that thing you are in the LoL client (see attachment 1)

Click "Play"

Then you have to login with your account (see attachment 2)

Then you are in the client.

Just click on the red button "Play" (see atachment 3)

Note that after you select play, you can choose to play a practice game or normal game. Normal games include matchmaking where the system automatically pairs you up with other players of the same ELO (read forum sticky for more information). Anyone can join a practice game you create, but you can choose to kick people.

If you're having the problem of when you push the start button and nothing happens, I had the same problem ): to fix that, all you need to do is uninstall then reinstall. Not sure about your question though, otherwise follow what Zafrada said