When you first start the game, you won't be able to cross to the other side of the river. This page explains how to be able to cross it.
Donate Bugs and Fish
Before you can cross the river on your island, you will need to get the Vaulting Pole. To get it, you first need to build the museum. To build it, you need to donate five creatures (bugs/fish) to Tom Nook in the Resident Services tent. After that, he will stop taking donations, and he'll give you the Blathers Tent Marker.
Build the Museum
Build the Blathers Tent Marker somewhere on your Island. The next day, the museum will be built.
Get the Vaulting Pole
Once the museum has been built, go in and talk to Blathers. He will give you a shovel, as well as the Vaulting Pole, which you can use to go across the river.