On our visit to Conwy we drove up from London. The trip was estimated to take around 4hr 15min, but took us about 5hr 30min, due to lots of traffic and road work.
We decided to drive, as having a car is the easiest way to explore Wales. Although there is a train network and various bus routes, having a car made it quick and easy for us to get to those off the beaten track places without any problem. For the most part, we found the roads in Wales to be in excellent condition and well signposted.
Another way to get to Conwy is by train. Virgin trains have a service that leaves from London Euston, via Chester to Llandudno Junction (the trip takes around 3.5hrs). From here you can change to the local train or catch a bus.
From the tourist office in Caernarfon I got the timetable for the local buses leaving Caernarfon, which was good to have as I was making daytrips to other places in northern Wales. Buses in Caernarfon all leave from Penllyn in the town centre.
Bus number 5 that runs between Caernarfon and Lludadno passes both Bangor and Conwy. I went to Conwy and it is a journey that takes 1h and 10 minutes. A return ticket was £5 (March 2012). Arriving to Conwy the bus stop is close to the castle and when returning to Caernarfon the bus stop is near the railway station. The buses run every 15 minutes.
As long as the weather is fine, there is a lovely little open top bus tour, run by the Citysightseeing Company, which will take you from Conwy to Llandudno past oodles of tourist attractions such as the Great Orme tramway in Llandudno and Llandudno Museum. It will also give you spectacular views of Conwy Castle and the surrounding Countryside. The ticket is valid for 24 hours and you can hop on and off the bus as many times as you want, whenever you want.
Conwy is a short bus ride from Llandudno. You can take the 5, 9 and their X varieties from Llandudno and be there in about 20 minutes. A return costs about 3-4 pounds. You can also get the train, although you'll probably have to change at Llandudno Junction. It will still probably be quicker, due to the fact that the bus meanders through the suburbs of Llandudno, stopping every few hundred meters.
You can also walk, if you've got the legs for it. It's about 3-4 kilometers to Conwy from Llandudno, and the views across the estuary to Snowdonia are fantastic.