For insects, there are a lot of different traps based on the nature of the insects being sought. There's a pretty good list here:

https://en.wikipedia.org/wiki/Insect_trap

As I understand the sampling, you pretty much just put the traps out and count up how many insects you get of each type.

Purely anecdotally, I feel like the number of butterflies is way down from the level you'd see during, say, the 90s. It seems like I just don't see as many different types any more and even the more common types feel vastly reduced in number. I used to see monarchs and sulfers and painted ladies and red admirals and cabbage butterflies all the time, now I barely see a few cabbage butterflies or sulfers and occasionally a painted lady or two. I never get to see the less common things like buckeyes any more, which is a pity given that they're my favorite.