I don't think there is one site for UK/RoI flights. It is sensible to book direct with the airline.
On many UK routes that is likely to be Easyjet or BA, on flights to the RoI it is more likely to be Ryanair or Aer Lingus/Emerald Airways.
On some UK routes Loganair may be the carrier.
Look at the individual airport website to see who operates the links you want. Almost all airports in the UK at least, and most in the RoI will have good transit links, but don't get hung up on that having to be by rail.
For instance Edinburgh is not rail connected, but has an excellent tram link and frequent dedicated express bus links into the city. Likewise Glasgow has an excellent bus link, but no rail.