Rules
The rules of Badminton are really very simple and certainly seem simpler on court than they read below.  You will get the idea after your first few rallies so don't worry about not knowing enough when you start.

Singles

In singles the court is long and narrow with the side tram lines never being used. All games are to 21 points.  A player/team scores a point when they win a rally regardless of who served. When a rally is won you also win the right to serve the next rally.

The serve is always diagonal from alternating sides, has to go over the service line and uses the back line as displayed. 

Serve 1 is out and serve 2 is in
Doubles

The Court Area for Doubles is long and wide.  Serving is always diagonal and must go over the service line.  Each time your side wins a rally you gain a point and the serve goes to the person who is on the right if the score is even and left if the score is odd.  You keep serving whenever you win on your own serve.  When the serving side loses a rally the serve passes to the opposition.  When they lose a rally the serve passes back to your side but the other player on your side.

In doubles the service court is shorter as shown on the diagram. After the serve, play uses the whole court.

Service 1 is in Service 2 is out.