Bug Catching is pretty similar to Fishing, but instead of the fishing rod you need to use your bug net. Bugs usually reside among trees. To catch them, slowly approach the tree and swing your net. Dragonflies and butterflies though can be found flying in the sky at different speeds. You have to run up to them quickly and catch them before they leave. Some are even found on trash, waste and flowers. These types are easy to catch and just get close to them and catch them. Finally, take note that bugs are easier to find when it is raining or when your gates are opened.
Caught Bugs can be donated to The Museum like Fishes, and it is best advised to donate the first kinds as well. Netting all kinds of bugs in the game rewards you the Golden Net, which is much larger than your regular net.