You are absolutely not too old to start any sport at the age of 15. That's positively young. I used to play badminton at club level and, whilst I had played other racket sports previously, I didn't start until I was in my mid 20s. Badminton clubs are struggling for members at the moment and any half decent one will absolutely welcome a teenager who wants to take the sport up. There will be badminton clubs near you and you probably don't need to do much more than Google and contact a few of them. There will also likely be one or more badminton leagues near you, and the league websites will have details for the individual clubs. Being 15 you'd be able to join a badminton youth club as well, but they are increasingly rare nowadays. In any event, you're old enough to join an adult club. Also look for beginner's badminton sessions that gyms with sports halls run. Those sorts of sessions are generally aimed at people who are new to the sport or want to play more casually, so would be a good place to start if you're worried about clubs being too competitive.