I would love some of your insights on when to check-- reason being that I rarely see "you should check that" as advice. As an example, I might bet out at an ace-high dry board without top pair, and with TPGK. But I would check a weak ace. These actions are also affected by position but thats a pretty consistent strategy of mine. This might be a rudimentary example but I still would love some insight into when it is actually appropriate to check.

There isn't a simple answer.

I'll give an example of an instance to check:

Hero is in position with a big draw and given action and board texture, V has a very strong range and is unlikely to fold.

This would be a good spot to check behind to try to catch the draw.

When you can not accomplish anything by betting

I'm going to lock this, but I'll give you a short piece of advice before I do.

When you do anything in a hand, you should be doing it for a reason.
There is a nice concept of the month involved with betting and the reasons why we bet. You can find it here. So in short, we should check when we are not accomplishing one of our goals by betting.

If we can get value from a range of callng hands that we are generally ahead of, we should bet.
If we can't we should check.

If you can think of two situations with similar hands, and you feel that you should bet one and check the other then post both of them, give your reasons and we will be happy to help you through your thought process.

But there are too many factors to give you a straight forward answer including:
Your image
The villain
Your hand
His range
Stac sizes
His tendencies
The board
Past history

In the very basic example that you gave:
We might check back the flop because the % of worse hands that can call us out of his entire range might be too small to make a value bet profitable. Or we might think that his calling range may wide significantly on the turn when we check the flop.

Good luck to you,
irtm