There's a bike route along the A55 from Bangor to Conwy - it may even go further but after Conwy you've got more choice and I turned south to Betwy-y-Coed last time I was around there. The bike route was mainly on tiny lanes, really narrow and lumpy but pretty much traffic free. It may be part of a national cycle route so check Sustrans website for more info - I found it by chance on the way out of Bangor a couple of years ago, so it could be things have changed or it might be better marked.
The A5 across Anglesy isn't so bad as most traffic, especially to/from the ferries uses the A55, of course. But the roads around the north of the island are a real pleasure if you're not in a hurry.