Is there any reason to keep bugs in my storage?
As noted, Harvest Moon: A New Beginning keeps track of how many critters you've caught and even breaks them further down into types caught, like "Cicadas Caught", "Bees Caught", etc.
However, unlike Fish, Bugs cannot be cooked, and the only place you can put them is in Storage. I'm pretty obsessive about collecting things, and I tend to like to keep things around unless I otherwise cannot store them. However, unlike the Animal Crossing series, it seems there's no separate place to collect Bugs and you must keep them in your house's Storage.
Given that my Storage is currently limited to 100 spaces and I've already caught 18 types of bugs running around during my first Spring day, I'm a little worried about running out of space. I'm thinking about just releasing/dropping them when it becomes necessary, but I'm worried I'll find out I can use them for something other than random statistics later on.
Is there any reason to keep Bugs around, or am I safe to let them go/sell them?
Bugs tend to be your main money-maker at the beginning of the game. There are some people that really like bugs (Aaron, especially), but that's no reason to store them, as they're easy enough to catch.
The only thing I would suggest is to keep about a half-dozen bees; you will be finding a use for them later on, and it will speed up development of that section if you have them on hand already.