For the cheapest get the bus from the airport to Liverpool south parkway station, then catch the train to Liverpool Central, then change at Liverpool central to the Wirral Line, and get the Chester train, then catch the Llandudno train at Chester, although it could take a while.
If you go with this it's worth asking how much the tickets are from Liverpool South Parkway to Llandudno. The train tickets from Wirral are usually around £14-17 (depending on the day), with a little research it might be cheaper getting an all zones saveaway ticket from Liverpool South Parkway then paying for the ticket to Llandudno at Chester. But thats the cheapest, not the quickest.