@JargonI'm going in order, by column (left to right).
Insect 1: Shake trees all year round, any time of day. I think you have five chances per day to catch this one.
Insect 2: Flies around your town between August and October, 5:00-7:00 p.m.
Insect 3: Grow a bush in your town, it'll be on it when it rains between April and September all day (best in May-June).
Insect 4: Shake a tree between October and February and one of these might hang from it...nab it quick, or else it'll climb back into it (don't worry, you can shake it again and they'll pop back out). Shake all of your trees to find it, including fruit trees.
Insect 5: Check your flowers between March-June from 8:00 a.m.-5:00 p.m. (also in October).
Insect 6: Chop down a tree, leave the stump. This guy is on it in June and September-November at various times in the morning and afternoon (never after 7:00 p.m., never before 4:00 a.m.)
Insect 7: On tree trunks, July-August. 11:00 p.m.-8:00 a.m.
Insect 8: Fish out some garbage or rot some turnips, then leave it out. You might find this little guy flying around it (year round, all hours).
Insect 9: Hit a rock with a shovel August-October, 7:00 p.m.-8:00 a.m.
Fish 1: Similar to the fish right above it. In the river year-round, 9:00 a.m.-4:00 p.m.
Fish 2: That's not coming to the river until March, so no worries.
Fish 3: He's in the pond part of the river, June-August, 9:00 a.m.-4:00 p.m.
Fish 4: Catch him at the waterfall (I've had success at both) March-June and September-November, 4:00-9:00 a.m. and 4:00-9:00 p.m.
Fish 5: He's out now through February in the river (and is huge), 4:00 p.m.-9:00 a.m.
Fish 6: All day in the river, but only in September.
Fish 7: This little river fish is basically out all day from June-mid-September.
Fish 8: This guy is in the ocean from now until February from 9:00 p.m.-4:00 a.m.
Fish 9: This larger fish (slightly bigger than a sea bass) is in the ocean from now until March all day long.