You will find that a new string instrument requires many tunings before the strings "settle" and stay in tune. The same applies to replacing a broken string. This is especially the case with nylon strings which stretch slightly under constant tension as the strings settle. Temperature and humidity changes or moving the harp will have a tendency to get the harp out of tune.
When the strings are eventually settled, they require only occasional tuning as needed. Keep the harp under even tension which will also help develop your own ear for tuning as time goes on. Harp strings are designed to resonate best at concert pitch, so keeping the harp consistently tuned to concert pitch will make the instrument sound better.
When moving from string to string as you tune your harp, be sure you have the tuning wrench on the correct tuning pin you want to adjust. If you continue to turn the tuning wrench but the string you're plucking is not changing pitch, you are probably turning the pin for the wrong string and are likely to break that string! More strings are broken accidentally during tuning (than by anything else) when people adjust one string while plucking (checking pitch) on a different string.
To avoid breaking strings while tuning use the "Up and Over" method for locating the correct pin for the string you are about to tune: Follow the string to be tuned up to its bridge pin and over to the tuning pin, and then look across the top of the harmonic curve (neck or pin block) to locate the correct pin for the string you are about to tune. Then put on the tuning wrench and adjust pitch while looking at your electronic tuner readout.
To avoid breaking a string, do not tune beyond the nominal pitch. This applies especially to wound stings as these are more prone to break if stretched beyond their limit. It will take repeated tunings until a new string "settles" and finally stays in tune. When you need to replace a string on a harp with threaded zither tuning pins you have to back out the tuning pin 2-4 full turns by turning the pin counter clock-wise. Use judgment as to how many turns it takes. When you are finished replacing a string you want to have the tuning pin hole lined up approximately with the groove on the bride pin.
Except for small lap harps (which generally have zither tuning pins), mid-size or larger harps today have tapered tuning pins which held in place by friction instead of a thread. The bore for a tapered tuning pin goes entirely through the pin block (aka Harmonic Curve) and it is tuned from the opposite side. Tapered pins are held in place by friction.
If a tuning pin ever slips (not holding the tuning), there is a simple solution: Just hit the pin gently on the end with a mallet or small hammer to tighten. This, obviously, applies only to harps with tapered pins. After you are done tuning your harp, DO NOT leave the tuning wrench on a tuning pin as the wrench could slip off the pin, fall the sound board and possibly causing damage.
Harp Tuning Wrench