What is a good way to determine the gravitational field strength somewhere (on Earth or in space)?

Answer

The simplest way is the experiment that is often performed in physics lessons in secondary education. You drop a ball from different heights and measure how long it takes before it falls to the ground. Gravity appears in this experiment as the acceleration with which the ball falls down. If the ball starts from rest, then the distance traveled is gt^2/2 (by t^2 I mean the square of the measured time). That g is then the ‘acceleration of gravity’, which is equal to GM/R^2, where G is the universal gravitational constant, M is the mass of the Earth, and R is our distance from the center.

Another method is to determine the period of a pendulum. If l is the length of the pendulum, the period is given by 2 pi times the square root of (l/g).

What you measure by dropping a ball is the acceleration the ball feels. It is of course only equal to the gravitational acceleration when no other acceleration acts on it. On Earth, the rotation provides a small additional acceleration, which contributes to the fact that the ‘effective gravitational acceleration’ differs slightly from place to place. The effect is important in a spacecraft. If it moves at a constant speed, then you measure the gravitational acceleration. But that is not possible close to Earth, because then the spacecraft is accelerated precisely by that gravitational acceleration. Spacecraft often move on orbits with an outward acceleration that compensates for the gravitational acceleration: the astronauts, if any, are then in a state of weightlessness. They then do not have the possibility to directly measure the gravitation.