If you connect to city water that does not fill your fresh water tank. To fill the fwt you need to put your hose in the tank fill and let it run in, takes a few minutes.
When you are on city water you do not want or need the pump to be on. The pump pumps water from the fwt. If the fwt is empty you should leave the pump off.
You would have to run quite a bit of water to get your grey tank to 2/3 full, which is 20 gallons. Probably that would take around 10 minutes running a faucet.
One more thing, don''t depend too much on the little level lights, they are notorious for being inaccurate. You can see the water level in the fwt by looking at it, its translucent. Best bet with the gray and black tanks is to start with them empty before setting up camp, then dump before you leave.