Brecon Beacons National Park

The main sight in Tintern is the Abbey. It is worth visiting, but it is a ruin. Caerphilly and Coch castles are about an hour drive from Tintern and both are worth visiting. If you go to Caerphilly castle the parking lot on the backside of the castle, along Crescent Road, is less busy than the small lot directly in front of the castle.

Near Tintern there are a number of castles that are free to enter including, White, Skenfrith, and Grosmont. Raglan [edited] and Chepstow castles are nearby, but there is an admission fee. Monmouth and its Monnow bridge is nice to walk around for about an hour. You can park for free, for an hour, at the Waitrose grocery store. Just get a ticket and put it in your vehicle window. Also nearby are Harold’s Stones; three Neolithic standing stones in Trelleck. Some of the roads in the Tintern area are very narrow.

Edited: depending on which and how many places you decide to visit, consider looking into the CADW pass that allows entrance into the CADW properties. It might make more financial sense.

Tintern Abbey is probably top of the list and definitely the closest.

https://cadw.gov.wales/visit/places-to-visit/tintern-abbey

Pictures

http://www.castlewales.com/tintern.html

While I like Caerphilly Castle my other suggestion would be Raglan Castle which is really impressive and a 30 minute drive.

https://cadw.gov.wales/visit/places-to-visit/raglan-castle

Have a look here for lots of pictures.

http://www.castlewales.com/rag_tour.html

I'd then follow this with a visit to Tretower Court and Castle, 30 min drive from Raglan.

https://cadw.gov.wales/visit/places-to-visit/tretower-court-and-castle
http://www.castlewales.com/tretwr.html

This takes you into open countryside and if it is a sunny day will be a much nicer drive than to Caerphilly.

Thanks so much. Very helpful information!