Manchester Airport: Use the airport rail station, not an expensive private transfer: take a train toward Llandudno Junction (some services are direct; others require a change, commonly at Chester). Allow roughly 2 hr 15 min–3 hr to Llandudno Junction, then a 5-minute taxi or a one-stop local train to Conwy. Advance rail fares are often about £25–45; the final taxi is roughly £8–12.
London Euston: The sensible rail gateway from London is Llandudno Junction: direct services take about 3 hours, then Conwy is a 5-minute taxi ride or a short onward train journey. Book an Advance fare rather than buying on the day; fares vary sharply, but £30–70 is a realistic advance range.
Chester railway station: A useful interchange for travellers coming from much of England. Take the North Wales Coast train toward Holyhead; Conwy is about 55–70 minutes away, with Llandudno Junction also a reliable fallback for the last short hop. Expect roughly £10–25 depending on ticket type and timing.
Conwy railway station: The tiny central station is on the North Wales Coast line and is the best arrival point when your train calls there: the castle, quay and walled core are a few minutes on foot. Do not assume every service stops here; Llandudno Junction is the more robust connection point and is only about 5 minutes away by taxi.