Now it's time for the more interactive part of the module. Instead of using text, I'll instead provide video examples in order to give you a much better idea of what you should do.
Top tip: Use the air/ground brake (X by default) to reduce the speed at any time you need it.
While there's no real checklist to follow, sit down in your jet and check everything before moving out. Follow this checklist if you want some guidelines:
There's no real procedures in Arma however to ensure your plane doesn't get damaged and you keep it fairly immersive, keep the speed under control and follow the taxiway lines where possible. You take them to the runway you're going to be taking off. If there's planes taking off ahead of you, you should ensure you're leaving enough space (ideally they'll have the runway to themselves). If a plane is landing ahead of you, you should wait outside of the runway.
Take-off checklist:
https://www.youtube.com/watch?v=QrTkyCMsov8
Carrier take-offs are a bit simpler as they just require you to park on top of one of the catapults, and an action will appear to launch your plane off. They don't require you to do anything for the actual launch.
https://www.youtube.com/watch?v=gVVurovLXOo
No video is required for this, but you should just put your plane in a position you're comfortable scouting from. For ground reconnaissance, you should be going fairly slow so you have sufficient time to observe the area before having to correct your plane's trajectory. Alternatively, you can attempt to fly your plane in a circle (doesn't matter whether it's in the clockwise/anti-clockwise direction).
For air patrols, you should keep your speed at a manageable level (not too fast to ensure you have optimal manuverability, but fast enough to ensure you can react to any threats) and have your plane at a higher altitude preferably to have a position advantage over any oncoming threats. Scan the area in all directions with your radar systems to ensure no aircraft can get the jump on you.
Landing checklist:
https://www.youtube.com/watch?v=8ch5BcYM1B0
Carrier landings are more difficult as they require you to be more precise in order to land safely. You will most likely need a plane with VTOL capabilities or a tailhook in order to make this work however it is possible to land without those (I don't recommend it though ).
https://www.youtube.com/watch?v=Q-U_FLSi7xY
Before atttempting to go out on a mission, you should be able to do most if not all of these things consistently. Ensure you practice enough before any mission whether it's a singleplayer or multiplayer scenario.