The closest Airports to Mumbles are Cardiff International Airport, Bristol Airport, London Heathrow, Birmingham and London Gatwick in that order. The ones nearest I would imagine would be the most convenient but probably the most expensive and flight times could be rather restricted as compared to Heathrow or Gatwick. Heathrow and Birmingham are about a 2.5 hour drive, Gatwick is a 3 hour drive from Mumbles. Trains are a possibility from either of the above airports but apart from Heathrow and Gatwick a Taxi would be required to take you to the nearest Train Station which could prove expensive for what it's worth. Therefore, if I was making the trip from the U.S I would probably fly to either Heathrow, Birmingham or Gatwick and then hire a car. As for visiting Ireland it really depends on how far you want to travel when you get there. A weekend in Dublin is fantastic but I'm sure you wouldn't want to miss out on the rest of the beautiful countryside and scenery especially around Cork and my favorite place, Kinsale (see attachment)