Moved from the Creative Cloud to the InDesign forum. They will be able to help you here.
wondering if there was a generic type of font that I can use for certain languages that have special characters. Or if InDesign has a function for this?
Not really. Any Adobe font that has the "Pro" designation - like the Myriad and Minion Pro families that come with InDesign - has a full complement of glyphs with coverage for Western European languages like French and German. Some (but not all!) Pro fonts also have Cyrillic coverage for Russian, and most (but not all!) such Cyrillic coverage includes Ukrainian. I've never had a problem typesetting Polish with Adobe "Pro" fonts. But the question is, where does Europe end? Is it Ukraine? Will you ever need to do Bulgarian or, let's see here, Serbian? Probably not, but if you do, you will need fonts with very extensive coverage of less-used Cyrillic glyphs.
What InDesign does have is "missing glyph protection" - it'll tell you that you are using a font that doesn't have the glyphs you need, if you're using a sufficiently recent version of InDesign.
If you're having the labels translated, you may want to talk to your translation providers about typesetting the labels for you. It's absurdly easy to mess it up if you're not already a localization pro.