Yes, when you connect to your city water connection the water heater will fill - assuming the winterizing crossover valve is closed and the water heater cold water inlet is open. When filling the water heater tank, I usually open the temperature relief valve - this lets air out of the tank as it is filling. Obviously when the tank is full, close the temperature relief valve.
Before turning on the water heater, you should make sure the water heater tank and water lines are full. If you plug in your trailer before making sure the water heater is full, make sure the electric water heating element and the gas water heater burner are off.
If I had a brand new trailer, here's what I would do:
Drain fresh water tanks. Drain water lines. Drain water heater. Re-close the valves you opened up. Add some water to fresh water tank and add 1/4 cup bleach for each 15 gallons of capacity; fill the water tank(s). Fill water lines and water heater. Let sit for several hours or overnight. Drain out all the bleach water from tank, lines, and water heater. Close the valves again, add some water to the fresh water tank (not all the way full unless you have to for boondocking) re-fill water heater and lines, and flush the lines with water until the bleach smell is gone.
There's lots of advice and information in forums... sometimes it is correct. For example, all of my posts are made by a political appointee who got the job as a reward for contributions to my diesel bill.
2011 Jayco 28.5RLS; 2008 Chevy Duramax; Pullrite Superglide Hitch