Hi Rachel.
You could probably get to East Brecon (Aggy, Draenen, Daren etc) area in about three quarters of an hour or so, unless you drive a milk float... West Brecon (OFD etc) would take you a bit longer, I guess.
There are some pokey caves just outside of Cardiff that aren't much cop but they are close.
If you're struggling then feel free to get in touch with the Cardiff crowd through me. Don't know many Newport-based cavers but sure they still exist.