To move around, just tap the screen. Your character will walk to the location you tapped.
You can also drag your finger across the screen, and your character will follow along.
Additionally, tapping the screen is how you perform many other actions, such as talking to other characters, purchasing furniture or clothing from shops, and picking up items (such as fruit or shells) from the ground. Fishing, catching bugs, and interacting with furniture (both inside your camper and outside) are also controlled by simply tapping the screen.