Taking our new-to-us 5er out for the first time!

Depending on whether your HW heater is full or empty the water pump will run for a few minutes to fill the tank. Once the HW tank is full and air is purged out of the lines the pump should shut off. If not, start checking for leaks. Mo leaks? Then it is possible the check valve in the pump is bad.
How are the tires? More than 5 years old it is time to replace, 3 years if they are ST tires.