Copper Mountain is an amazing place to visit - like a lunar landscape. It includes a museum in Amlwch. Also in Amlwch is a fascinating cove (called Amlwch Creek) with a precarious walk hacked into the rock - you can get down to the sand when the tide is out. Amazing views. The port is also nice.
Lovely beaches to visit at Treaddur Bay, Lligwy, Cemaes and Benllech - many more of course but those are good. Oh and Cable Bay - but parking can be cramped/pricey. Fabulous walk along the headland though to the burial chamber.
At Penrhos there is a nature reserve with beautiful walks, make sure to get to the end where there is a secluded beach that looks like the Caribbean. Good cafe too (at the car park)
Plas Newydd is worth a visit if you have NT membership.
Menai Bridge is worth a visit. We park by the Jade Village Chinese on the outskirts then take a walk down to the very picturesque St Tysilios church (on its own island), where you have great views of both bridges. Then you can walk along the Belgian Promenade under Menai bridge into the town, eventually circling back to the car park.
Beaumaris is totally worth a visit, including the castle. If you carry on driving out past Beaumaris to Penmon you come to the Priory with its amazing dovecot and ancient well. You can then take the toll road out to the lighthouse (well worth it) the beach is pebbly, but the views are good and there is a cafe. If you are feeling very adventurous then there are fast rib rides from Beaumaris out to Puffin Island to see seals and puffins (there are slower more sedate boats too).
At Holyhead, although the town is nothing special, you do have the country park with walks up the mountain, plus the breakwater itself which you can walk along and is brilliant for crabbing.
Other walks include the Dingle in Llangefni, which is a boardwalk in places - you can explore the old railway and walk through the 'gorge' to get to the reservoir at the end. There are red squirrels in the woods.
Look out for the Giants Stepping stones on the river Braint near Newborough. Parking is tricky though - don't drive down the access track, walk from further. Of course Newborough also has an incredible beach, and a fabulous walk to Llanddwyn - which you can cross to when the tide is out (worth doing so check the timings). Also red squirrels in the woods behind.
This is only a small number of things to do on the island. I haven't even started on what else is on the mainland. We'll be on the island next Saturday - I can't wait!