If you're always looking for a fight, consider yourself disputatious. However, disputatious is more appropriate for the head of the debate team rather than the playground bully.
Disputatious is a somewhat old-fashioned term now. In every day speech we'd probably use a dozen more colorful words, many not printable here. The one area where it does live on is in the legal sphere — two battling parties are often said to be disputatious. And, in fact, an extreme form of being disputatious often involves frequently taking people to court, or being litigious. Only lawyers love disputatious, litigious people.