PF is kickass!
Congrats on diving into a serious challenge
Carrier takeoff is no joke. Make sure you have flaps set to takeoff, and your engine revved wide open... the trick here is to make sure that you map your chock-release to a key. It's not mapped by default. Alternatlively, you can make sure you hit autopilot right as the mission begins. For whatever reason, if you don't do it right away, the AP can't handle it. You will also have to touch the rudder a bit to counteract the ship's heaving.
Anyway, flaps at takeoff, engine wide open, and release chocks. It's not a rocket, so let it launch off the deck and gently pull up and retract gear. Then retract flaps a notch, let airspeed build a few seconds (depends on the plane you are flying) and then retract them again all the way. From there you are flying
There may be a way to play this game with keyboard/mouse combo, but I wouldn't expect to be able to do much of anything except crash a lot. A decent joystick is critical to this type of game.
Landing is again a real challenge. Real life standard military procedure is to pass down the length of the runway at low speed, then circle around to make an approach at the runway with gear down, landing flaps deployed. The real important part is to pay very close attention to your climb indicator (again, where it is on your dash depends on the plane you are flying) and keep make your approach so that the indicator is very close to level when you make your flare (Flare is when you pull your nose level just before the landing gear touch the ground). The best thing I can tell you here is to practice a lot. Keep in mind, that real life pilots practice touch and goes a LOT in order to keep their landing skills up to snuff, because it ain't an easy thing to do
All of this won't mean much to you trying to use a keyboard/mouse combo. Flying that way would be similar to playing an FPS with just the keyboard. It just don't work very well. You will be very amazed at the difference a decent joystick makes in your flying
If you ever want to wing with me, I'd be happy to fly with you