Don't listen to ilove, Anglesey is great for a visit (you just might not want to live there )
Newborough is great, loads of sand and dunes - especially recommended is the walk out to Llanddwyn island. Newborough warrens and pine forest is nice for a walk too, alledgedly there are red squirrels here.
Cable Bay is a sweet, pituresque little beach, but can get very crowded. There is a nice walk along the headland to the burial chambers.
The town beach at Rhosneigr is good if you like watching wind-sports.
South Stack is a lovely walk, though you can't often cross over to the lighthouse itself. The cliffs and seabirds are amazing as are the views. The rest of Holyhead island is probably worth avoiding though !! Apart from Penrhos nature reserve.
Along the north coast there are some very unspoilt sandy beaches, but we don't often venture up that far.
Red Wharf Bay is worth a visit, not very beachy in itself (being a bay), but nice scenery, there is a nice pub called The Ship Inn. There are other beaches in the vicinity - Benllech is good for tourist ameneties (but parking can be tricky)
The walk up to Penmon Priary is lovely, well worth it, there are no beaches this end (just pebbles) but it's a nice coast walk to look out at Ynys Seiriol (Puffin Island)
Beaumaris is nice for a visit, especially the castle.
There are loads of other small beaches, some of them quite hard to reach (no road access), but worth the effort. Look around 4-mile Bridge or Treaddur Bay. The Anchorage pub is meant to do nice food, but gets busy.
Other places - Pili Palas is nice for the little ones and has a soft play place. The Sea Zoo can get crowded, but little ones love it, they also have a nice playground which you don't have to go to the zoo to use. Not sure if Henblas Country Park is still open, but it used to have fun stuff for kids going on. Plas Newydd is also worth visiting, even for little ones.
As for eating out, it can be a bit hit and miss, nowhere is outstanding in terms of pub grub. There are some fancy restaurants, but we haven't tried many ourselves.
Aware I'm raving now, but I'd recommend one other place on the mainland for kids - GreenWood Forest Park is a great day out.