Your first flight

For an introductory flight for a novice orbiter user, a good start is the Delta Glider docked to the ISS. This way, you can explore orbital mechanics, use of RCS thrusters, and docking with the most user-friendly standard craft in orbiter in a forgiving space environment.

Task 1: Undocking

To undock, you need follow steps:

Make sure your RCS thrusters are set to translation/linear mode.

Look towards the bottom of your HUD (alt+down), where it says either RCS:LIN or RCS:ROT.
If it says RCS:LIN, do nothing. If it says RCS:ROT, press the numpad "/" key.

Separate yourself From the ISS

Press Ctrl+D to undock.
Press the numpad "9" key to thrust your ship away from the ISS.
WARNING: Only press 9 for a moment. If you use too much thrust, you will degrade your orbit and reenter the atmosphere.

Congratulations! You have successfully undocked from the ISS!

Task 2: Making a larger orbit

You may want to look at our beautiful home-world from a greater distance (i.e. higher orbit.) There are three main steps to this.

Set your MFD

On your MFDs (the screens on your instrument panel) make sure one is set to Orbit. if neither are set to orbit, choose one MFD and press the SEL button on the bottom. Press the button next to Orbit.
Press the REF button next to the display. Use the arrow keys to navigate to earth on the menu.
Press the DIST button until you see that two of the readings read ApA and PeA. These should each have some number from 200k to 300k

Raise your Apoapsis altitude

Note: your apoapsis (aka apogee) is the highest point on your orbit. your periapsis (aka perigee) is the lowest point on your orbit.
Look at your PeT (time until periapsis) reading on the orbit MFD. this reads in seconds, with metric prefix abbreviations. For example, PeT=1.287k means 1287 seconds until you are at your periapsis.
When your display shows PeT=60, press the "PRO GRADE" button next to your HUD. (Look back up at your HUD with alt+up.) It should light up, and your ship should start rotating towards your direction of motion (represented by a circle with a "+" in it; you may have to press "HUD" on the Orbit MFD or "OBT" on the HUD for this to appear).
Note: You may use time acceleration to get to PeT=60 by pressing "T". Stop time acceleration with "R". Be careful not to overshoot your target.
When your display shows PeT=0, fire your main engine by holding "+" until your ApA (apoapsis altitude) reading is ~500k, then stop.

Raise your Periapsis altitude

Look at your ApT (Time until apoapsis). When your display shows ApT=60, make sure you are facing prograde.
WARNING: The prograde button activates an autopilot. AUTOPILOTS CANNOT HANDLE TIME ACCELERATION ABOVE 10x. Turn off prograde autopilot by pressing PROGRD again before using time acceleration, then press PROGRD a third time to orient yourself at ApT=60.
When your display shows ApT=0, fire your main engine until your PeA (periapsis altitude) reads ~450k, then kill thrust.

Congratulations! you have performed your first orbital maneuver! Now you can take a well-earned break.