Bristol Airport is the closest airport to Bath (only twenty miles away). You can fly to Bristol from most major cities, including Edinburgh, Paris, Brussels, Dublin, Amsterdam and New York.
To travel from Bristol Airport to Bath, you can either take an Airport Bus to Bristol Temple Meads railway station and then take a train to Bath Spa or Chippenham railway stations, or you can take a direct Airport Bus to Bath city centre. For Newton Park it may be quicker to take a taxi directly from the airport. The current quoted price is £40.
Alternatively, you can fly to London Heathrow, London Stansted, or London Gatwick airports and take a train to London Paddington railway station, from where you can travel onwards to Bath Spa or Chippenham railway stations.