So for the most part you'll only ever measure your instantaneous speed relative to the air mass in which you're flying (Indicated Airspeed/True Airspeed). This is most useful for staying in the air, as it's the speed at which air is moving over your control and lifting surfaces. This is done with a pitot-static system (pic related) which combines measurement of ram air pressure (force from forward speed of air ramming into a tube) and stagnation pressure (ambient pressure from altitude).
Ground speed needs to be calculated by essentially timing how long it takes to traverse between two points on the ground. This is not straightforward and is the main reason that most planes that had a mission dependent on this sort of thing had a dedicated navigator crewman. Particularly if it had to be done for periods away from the ability to make observations, such as at night or for extended periods over water.
Observation points might be:
>Large cities, particularly with distinctive buildings such as cathederals
>Natural features or formations, mountains, lakes, rivers
>If at night, sextant readings of the stars
>radio beacon direction finding (which is probably worth its own post. Google 'battle of the beams')
