To add roads, tap to open the Shop, then select the tab with the stone path. Once in the road tab, you can swipe left to find the road you would like to add. Tap the road to select it, and if necessary to purchase it, then drag it onto an open area in the world. This takes you into Edit Mode, where you can add or remove multiple road sections at the same time.
When you are finished placing roads, tap the Done button to exit Edit Mode. You can re-enter Edit Mode by tapping and holding on any section of an existing road.