In an electric circuit when electrons move from low to high potential, will they gain energy or lose energy?

They will gain energy.  The general flow of electrons in an electrical circuit is from the low potential to the high potential.  To do this, they can't help but absorb some energy as they shift positions in the various stages of the circuit.  At the point where the electrons start their journey through the circuit, the potential is low.  The energy accompanying these electrons is low as well.  As they travel through the conducting material of the circuit, the electrons pick up energy as they travel.  They finally max out at the top of the electrical potential spectrum as they arrive back at the power source in the high potential terminal.  Electrons are negative in charge, so they flow from the negative terminal of the battery, which is usually on the bottom, through the circuit, and on their way to the positive terminal, which is on top.  While they accomplish this journey, they accomplish some sort of work, such as heating a filament in a light bulb, which causes it to glow and give off light.