Strafing runs are how you're supposed to handle fast fighters. Climb high, and then "boom and zoom" your opponents.
Dogfighting is trickier, as you get stuck in a low energy state and you need to be able to outturn your enemy and gain some speed back. Lucky for you, Japan is the best nation to practice this.
Outside of a few planes (the Ki-61s and J2Ms don't turn very well) just about every Japanese fighter, especially the Zero, turn amazingly well. Get the Zero and practice dogfighting in it. Trust me, it's a blast once you get the hang of it.
And remember to click "F" and use combat flaps to turn faster!