A few things I've learned about aircraft in KSP for those research missions. I'm posting this 'cuz this is a reasonably good summary of what I've waded through to get some good info on how to do things.
Construction:
Simpler is better.
Turn on Center of Mass and Center of Lift. In order to build a plane that doesn't spin out of control, Center of Lift must be BEHIND Center of Mass and should ideally be left/right aligned with the center of mass. Using Mirror mode when adding parts is great because whatever you do on one side goes on the other.
Turn off Cockpit Torque (Reaction Wheels: Disabled) and empty your cockpit of monopropellant. It's basically useless as control surfaces adjust your heading in planes UNTIL YOU GET TO SPACE which you can't do until MUCH later.
Your aircraft will be nearly useless at the beginning (Aviation, L4) EXCEPT to run around to all of the buildings and terrains nearby and do science, which should be enough to buy Aerodynamics (L5).
When you get Aerodynamics (L5) and the Wheesley engine you can build a half-decent plane. Be warned: the initial fixed landing gear is both annoying in that it creates a ton of air resistance and is basically less reliable than building struts out of toothpicks. I simply toss together a Mk1 cockpit, 2 tanks of fuel (2x400=800), swept wings, the wheesley, a radial air intake, some landing gear, forward canards (tail fins mounted horizontally), a tail fin, some elevons, and one of each of the mission science toys I have (thermometer, barometer, seismometer). Warning: a Wheesley is only efficient at near-sea-level. The higher you go, the less thrust it produces. Warning 2: The
When you get Landing (L5) you can replace your toothpicks with foldable Small Landing Gear which should nearly eliminate the problem of breaking landing gear when landing if you come down reasonably gently, and it folds up to reduce air resistance when flying. Set Springs and Dampening to 2 and your rear brakes to 200%.
With Supersonic Flight (L6) you can make 20km with a basic plane with a Mk1 cockpit, 4 tanks of fuel (4x400=1600), those idiotic rectangular wings with 2 elevons on each, a single aft-mounted afterburner engine, and 1 or 2 top-mounted Radial Air Intakes. Despite is high fuel usage, afterburn is VERY efficient at moderate altitudes (5-15km) and high speed (700-800mps), even more so than your Wheesley, because of its supersonic thrust multiplier: it gets more powerful for less fuel the faster you go, up to about 800mps. If you want to take on those "above X" contracts and don't want to use disposable booster rockets, consider this design. Warning: it does not have jet-like maneuverability above 15km, that is, trying to turn your plane around will be next to impossible as you will end up pointed perpendiular or opposite to your direction of motion, which in a plane is NOT GOOD.
With High Altitude Flight (L7) you can replace your Radial Air Intakes with Adjustable Ramp Air Intakes (Radial).
With Hypersonic Flight (L8) you get a RAMJET engine and Cooling Intakes and after that at L9 you get the Rapier.
Remember, more X is not always better. If you double your fuel, engines, and wings, you essentially have the same plane for twice the expense!
Flying:
Remember the speed at which your plane can lift off the ground, which you will need when landing.
Be gentle with turning, especially when near the ground.
If you can see texture when you look down, the ground is about 500m below you. Consider pulling up.
Landing requires you to (1) get near the ground, (2) slow down, (3) keep your plane pointed up about +1 or +2 degrees or so while you land, and (4) brake until it stops. (1) is easy. (2) means shut down your engines to the point where you're just barely keeping it flying and then cut them totally when you're just above the surface. If necessary, you may need to pull back on the stick hard to climb, and then push forward on the stick to fall, to get rid of some of that speed. (3) means you should not let your plane speed fall below the speed necessary to keep your plane flying until you're ready to land. If you do, your plane's nose will point down and you will be unable to get it back up before you crash. (4) is the hard part, particularly in Highland and Mountain areas which can be quite varied so your plane will bounce around and/or tip over and break its wings off.