From what I've read in this forum, it seems as though everyone agrees that ceramic tile would work in this environment. Is that correct?

But I'm not as sure if I want tile because it is so hard and cold. Another option which has appeared is vinyl planks or tiles. I've been to two flooring stores and one said to just do click-down and avoid glues, while at the other store they said just the opposite. So, is vinyl an option? If so, should it be glued or not? From what I've read on here, a vinyl sheet would be a no-no because of the concrete beneath would not be able to breathe. True?

And for either the vinyl or the tile, would we need to put anything on top of the cement floor?

Thanks in advance for your help.

Tile on concrete is cold indeed. You'll probably want to put area rugs on top.

Older concrete basement floors were often only a couple of inches thick with no gravel or vapour barriers underneath and no mesh reinforcement in the concrete. So you'll want to check for excessive moisture and definitely use an uncoupling membrane such as Ditra under ceramic tiles.