Without a car, your biggest problem is going to be transport. Out of season, buses run far less frequently (if at all) and may only run two or three times a day even to popular locations. Contrariwise, if you arrive in mid-season you get zillions of people crowding the place. So, I would recommend visiting at one end or the other of the peak holiday season: say May or June (school holidays start in July) or September (schools go back 1st week in September). This would also help with finding suitable accommodation.
Have fun, Bob.
P.S. Useful sites: Trains: http://www.arrivatrainswales.co.uk/ there are maps and timetables in there... Click on "Travelling With Us"
Buses: list of links to the various Welsh bus companies: http://www.bbc.co.uk/wales/southeast...es/buses.shtml