abergavenny nice market town with good walking nearby - the skirrid, sugar loaf, pen-y-fan & corn ddu, the blorenge but not as quiet as ystradfellte which has excellent walks around waterfalls, open moorland, lots of caves and is very quiet. the waterfalls (about 8 really nice ones) are fantastic for walking the dog although beware occasional steep drops.
both nice but would go for ystradfellte if you want quiet. gower for beach walks would be within striking distance also...rhosilli / worms head always a favourite
seeing your avatar it may be worth commenting on the caves a little more. one of the caves near ystradfellte is the (in)famous porth-y-ogof...over a dozen deaths in the last 40 odd years. also really good show caves nearby at dan-yr-ogof.
also several walk in silica mines in same area (dinas)