What does the suffix -unque mean?

It means any or ever. E.g. come is how and comunque means however or anyhow. Dove is where and dovunque translates to anywhere or wherever. Quale is what and qualunque means just any, -unque is not a word that can be used alone, but only a word ending. Anything or whatever is qualunque cosa, the word for thing is cosa. Chi is who and chiunque means whoever. I'm not just anyone, Non sono uno qualunque (not am one any).

You can also find -unque in other words, e.g. dunque. Here the prefix is just d and does not have any meaning by itself. In this case just learn that this word as a connector (conjuction in grammar jargon) means therefore, so.

What's the difference between fare la spesa and fare spese?

Shopping has no plural in English, but spesa has a plural: spese. Far(e) la spesa is grocery shopping, while far(e) spese/compere/acquisti is shopping for clothes etc.