Conwy's walled town is tiny — you can walk from one side to the other in ten minutes. The main sights are all on foot, and the town walls themselves serve as the best way to understand the layout: climb up at the mill gate and walk the circuit for a view over the rooftops.
For day trips into Snowdonia and the Conwy Valley, a car is far more useful than public transport, though the Conwy Valley railway line (from Llandudno Junction) will get you as far as Betws-y-Coed without one.