Are there certain grades who should be taking these? And which tests?

It depends on where you live.

In my state, children 7-16 must take a test each year.
We use the California Achievement Test

Some states do not require testing

Hope

Some people like to have their kids take them whether they're required by law or not. If you're one of those, tests are available for every grade.

In our state, it is required every other year, but the parents can choose from a variety of standardized tests.

Tara

In GA we are required to test in 3rd 6th and 9th. The laws vary greatly from state to state.

In FL there are no requirements to test. You may choose this option as one way of fulfilling the obligation of your "annual evaluation." This can also be done via consultation/eval with a licensed psychologist or FL licensed teacher, without any testing.