I installed a new pressure tank. I filled it, but it did not seem to fill to capacity. There is barely five gallons in there. After flushing the toilet once, the pump kicks in again. Is there anything I need to do to make sure the tank is installed correctly, and that it is working properly?

You failed to calibrate your bladder tank. To set the air pressure in a bladder tank, look at the cut-in pressure on your pressure switch. Now drain the line down with the pump off. When there is no more water coming out of the hose bib, set the pressure 2 lbs. below the cut-in pressure. Check it with a tire gage. If your pressure control is set to come on at 20 PSI and go off at 40 PSI then your gage should read 18 PSI. Hope this helps and thank you for rating my reply.