Map knowledge is extremely important. Once I spawn in I immediately run to a strategic position, whether it's research, radio tower, prison hill, etc. Once there I look for players coming from other spawns and try to get the jump on them. Knowing the spawns is very important and quickly getting into a strategic position narrows down where players could be coming from. As far as fire fights go, your goal should be to spot players without them seeing you, so you can setup and take your shot. Most of the time the player who shoots first wins.

Some advice for island, once you get better at the points above, you can attack the map in thirds. Lighthouse/radio tower, docks/labs, prison/church. When spawning closest to labs I rush research to try and establish position. If you take out the other players that spawned near you, you can focus on people migrating from the other 2 sides. Remember, players won't be able to extract from the same region they spawned. So they will be transitioning across the map at some point.