From Heathrow or Gatwick Airport, it may be possible to travel by National Express coach to Cardiff and thence to Brecon. Otherwise, it is best to take the train to Reading, then another train to Cardiff.
From Cardiff Airport there is a bus to Cardiff Central Station. At certain times this service is not very frequent.
From Manchester Airport there is a train to Manchester Piccadilly station then another train to Abergavenny.
From Bristol Airport there is a bus to Bristol Temple Meads station and then train to Abergavenny.
From Holyhead it is possible to travel to Abergavenny by train (sometimes involving a change at Crewe).