You have been offered sound advice by isn above. Dolgellau has a cycle trail that is in fact a former railway and is also right by the Coed-Y-Brennin Mountain Bike trails. Barmouth is also another possible place where you might like to stay. Both these towns are also a shorter drive up from Cardiff than trying to reach Conwy on the north coast. People on RS all seem to head to Conwy because RS says that it the place to go. That may be so if you don’t have a car but being as you are driving, you have the freedom to choose a wide choice of locations. Even for people without car, I would suggest that Porthmadog makes a good base for Snowdonia.
Be warned that some of the ‘A’ roads in the Snowdonia National Park can get very busy with traffic, they also have some very steep hills and are narrow. You would not want to be riding a bike on these roads with a load of cars wanting to overtake and when they do be breathing in the fumes.
Be sure to visit Harlech Castle & Portmeirion whilst in south Snowdonia. Conwy is not such a good base as it is so far north.
https://www.sustrans.org.uk/find-a-route-on-the-national-cycle-network/mawddach-trail-dolgellau/
https://www.beicsbrenin.co.uk