A music supervisor's job is multifaceted. You need to have a balanced and impartial music taste as you'll often be listening to tons of different styles and genres, and need to work with the director's/client's requests.
You need to be well organised and understand how to priortise properly too. Read up as much as you can about copyright law, ownership of music on an international level. Dig around on PROs like BMI and ASCAP, understand what it all means. Go to gigs, find new bands and artists. To get started in the industry, find people making short films and help them out. I got an internship, and from there ended up working there for a long time. You have to really want it, and just dive in and go to EVERY event you can, and meet anyone and everyone.