So I have created many curved text boxes by placing the text onto curves, and choosing Text... Put on Path...

I have series of circles and triangles that are enclosed, and I place these curved text boxes inside (with the curves turned off).

I had no idea I would not be able to just fill the backgrounds, but would need to click into each and every enclosed letter to fill it in with a color.

Is there a way to do this, to fill in a background with color, where there is curved text, and also that is enclosed inside another circle or triangle?

Hi.

You need to avoid the bucket tool for that.
Draw the filled area as a shape or path, and use the fill and stroke panel (Ctrl+Shift+F).
For that matter, adding a fill colour to the ellipse would do the trick.
Move it to the bottom for the right z-order (End).

If you have trouble with Lazurs's suggestion, it would be helpful for us if you provide at least a screenshot, to show us what's happening in the image. I don't follow your verbal description.