Carcassonne is not far from the Cathar castles:
Montsegur
Peyrepertuse
Queribus
Aguilar
Puivert, amongst others.
I'd second: Mirepoix, Camon ('prettiest town in France' apparently), Axat (walk the sentier du tren, short easy mountain walk). Near Puivert castle (which was in the weirdly fab Johnny Depp film The Ninth Gate), is a walk called Labrynthe Vert near a small town called Nebias. Beautiful interesting walk around a natural labyrinth.
Bugerach near Quillan is an interesting mountain formation, with a very swimmable lake and waterfall (Cascade Matthieux - excuse probable misspelling, I'm not French!).
We drive up from Spain, so enjoy the mountains and greenery rather than the beach, so can't comment there.
We enjoyed Foix and the Caves of Lombrives many years ago. There are several caves fitted about, so you may find one closer to Carcassonne.
The Orgues of Ille Sur Tet as a pp mentioned are stunning. We also liked Tautavel and Europe's oldest skeleton (400,000 years), although the museum was a little old fashioned, it was fun. There's a dinosaur museum in Esperanza, that whole area of France is full of pre-historical finds.
We've never driven past Carcassonne further into France as we stay the other side of Quillan. Maybe one day we'll rent a boat and go down the Canal du Midi.