I'm from North Wales and regularly cycle on the A55 between Llanfairfechan and Bangor without incident. You can avoid a lot of the A55 by taking the costal road, some of which runs alongside the A55 (past Conwy & a little along Abergwyngregyn). There is a Sustrans route to avoid the A55 section from Abergwyngregyn to Tal-y-Bont (Bangor). It's a single lane through route for local traffic only. They've resurfaced it now so it's a lot better, but you need to watch out for the cars, drivers of which have looks of horror on their faces when they come haring round the bend and realise there is a cyclist coming in the other direction (hence my preference for the A55 if I know there is no ferry traffic). There are sections of the A55 you can't cycle on anyway (e.g. Colwyn Bay - Conwy).
I've cycled parts of the A5. That is ok depending on what time of day and time of year you travel on it. It can get very, very busy. We're off to Bala camping this weekend, and rather than take the A5 we'll probably do the A4085 & A4212 instead.
I'd personally take the costal route - NCN 5 route. It's a lot flatter for a start.
Don't know if this site will be of help:
http://www.cyclingnorthwales.co.uk/index.html