There are heaps of bugs to catch (namely 40). This list will help you
to track them all down :)
Keep in mind that some bugs can be found in more than one season.

Season: All season | Spring | Summer | Autumn | Winter | Check List

Summer bugs: (June-August)
Banded Dragonfly
Brown Cicada
Cockroach
Common Butterfly
Common Dragonfly
Darner Dragonfly
Drone Beetle
Dynastid Beetle
Evening Cicada
Firefly
Flat Stag Beetle
Giant Beetle
Grasshopper
Jewel Beetle
Ladybug
Long Locust
Longhorn Beetle
Mantis
Mosquito
Mountain Beetle
Pondskater
Purple Butterfly
Robust Cicada
Saw Stag Beetle
Snail
Spider
Spotted Ladybug
Tiger Butterfly
Walker cicada
Yellow Butterfly
July-Aug
July-Aug
Mar-Nov
Mar-Oct
May-July
June-Aug
July-Aug
July-Aug
July-Aug
June
June-Aug
July-Aug
Aug-Sept
July-Aug
Mar-July, Oct
Aug-Nov
June-Aug
Aug-Sept
May-Sept
July-Aug
June-Sept
June-Aug
July-Aug
July-Aug
Apr-Sept
Apr-Sept
Mar-July, Oct
Apr-Sept
July-Sept
Mar-Oct
8am-5pm
8am-5pm
All day
8am-5pm
8am-5pm
8am-5pm
All day
7pm-8am
4am-8am,4pm-7pm
7pm-4am
7pm-8am
11pm-8am
8am-5pm
8am-4pm
8am-5pm
8am-5pm
8am-5pm
8am-5pm
8am-11pm
7pm-8am
8am-7pm
8am-5pm
8am-5pm
7pm-8am
All day
All day
8am-5pm
8am-5pm
8am-5pm
8am-5pm
Anywhere
on trees
on trees, flowers, attracted to turnips
Anywhere
Anywhere
Anywhere
on trees
on trees
on trees
near water
on trees
on trees
in grass
on trees
on flowers
in grass
Anywhere
on flowers
Anywhere
on trees
on the ponds
Anywhere
on trees
on trees
on flowers
in trees
on flowers
Anywhere
on trees
Anywhere
4,500 Bells
200 Bells
5 Bells
80 Bells
130 Bells
200 Bells
80 Bells
1,350 Bells
850 Bells
250 Bells
2,000 Bells
10,000 Bells
130 Bells
3,000 Bells
130 Bells
200 Bells
200 Bells
430 Bells
130 Bells
2,000 Bells
130 Bells
2,000 Bells
300 Bells
2,000 Bells
250 Bells
300 Bells
200 Bells
200 Bells
400 Bells
80 Bells