I'm seeing a cherry bush that has all the flowers up high (out of reach), and a current bush that doesn't seem to really be producing new growth. Should I cut them or any other bushes to the ground?
That's a bit drastic. Pruning is an art and a science, and cutting things to the ground tends to kill a lot of the ones that are not outright weeds.
Step one, sort out exactly what you have in the wide range of things referred to as cherries, as best you can.
Step two, look up when and how to prune that variety/species for the desired results.
Lacking detail, rules of thumb for "drastic" pruning (which vary, as thumbs do) are on the one hand to leave at least a third of the growth, or on the other to cut only a third of the growth each season. i.e. cut off 2/3 or 1/3 of present growth this year, next year, looking at the growth you have next year to start from, cut off 2/3 or 1/3 of that, repeat until you get where you want it to be. But it's best to have an idea what the specific plant is, as different plants respond differently, and some are more or less easily offended by drastic pruning - also the best timing of any pruning varies by what result you are trying to achieve - early spring is best for encouraging new growth, summer is better for size control, and it's generally unwise to prune from late summer through fall.
Grapes are an exception, preferring to be pruned in the dead of winter.
I don't have experience pruning cherry bushes, and I don't have a recommendation about what the best thing to do is with cherry bushes and currants. However, I can tell you what will probably happen if you prune your currants like that, based on my experience with pruning currants.
If you keep at least a couple feet of the plant (instead of coppicing it), and prune in spring then don't expect much of any fruit the first year, but the next year you'll probably get plenty of fruit, and a big plant again. I'm guessing they would survive coppicing, and behave similarly, but it would be a lot riskier.
Fall is the best time to prune them, in my opinion; in the spring, you risk pruning off the branches where flowers have decided they're going to appear. If you prune them in the summer, or when it's hot, they can die.
This whole answer is assuming your plants are not young. Pruning younger plants is probably a different animal.
I would expect cherry bushes to be more sensitive to pruning than currants. I wouldn't coppice them, especially if they're grafted. We've had cherry bushes, but they were never vigorous enough that we pruned them, IIRC.
Blackcurrants are vigorous and multiply pretty quickly, in my area. They seem to grow about five feet a year, when pruned.