Most types of buildings require connection by road to your town hall: residential, production, goods, military, and cultural all do. This isn't anything to do with moving them, other than it will need a road for it to be functional in its new location.
You can move a building by opening up the build menu (hammer symbol). In the browser version, you get 3 icons in the top center of your screen. The middle one, with 4 arrows, is the move icon. Click it, click on the building you want to move to grab it, and place it in its new location. I'm not sure if you get those icons in the mobile version, or where they appear.