In reply to ceri:
The coastal bits are almost universally lovely, rugged on the western side and with lots of hidden coves on the east. The Straits can feel a bit crowded in places (comparatively) but it's worth it for Beaumaris and the views back to Snowdonia. Newborough and the stretch up to Rhoscolyn is possibly the quieter side, but there are some lonely stretches on the northern coast. The area around Wylfa is going to see some signifcant disruption in the coming years with the work on the power station.
It's small enough that all of it is accessible by car without too much effort, and there are quite a few 'locals' spots that can be quiet even in the middle of summer. It's a bit like a smaller, quieter Cornwall.
There are good local amenities at either end of the island (Menai and Holyhead) and 'big' shops in Bangor.
Strongly recommend getting to grips with Welsh as it's the first language on most of the island.