I can't help with campsites as we tend to go for stealth overnights (quite tricky on Anglesey) or else day trips from home, but there's plenty to go at on Anglesey, especially if you like coastal walking and scenery.
I agree with DuxDeluxe, the place we return to time and time again is Newborough / Llanddwyn / Maltraeth - beach, dunes, forest, rocks and geology, wildlife, it has so much and lots of scope for varied walks. If you can time the tide right try crossing the bay beyond the dunes out to Newborough Point.
But there's so much other fantastic coast. Holyhead Mountain and North Stack and South Stack are spectacular, and the RSPB have a visitor centre there. Other favourites of ours are the dunes and beach at Aberfraw and the section of coast up to Rhosneigr. Church Bay to Carmel Head in the north is good, and Cemlyn Bay which you might be able to time for the breeding tern colony. The coastal path from Red Wharf Bay towards Penmon and Puffin Island is interesting too.
And as An-j says, the old copper mine at Parys Mountain is quite amazing, the colours are just incredible.
Speaking of puffins, chances of seeing them vary, but there are boat trips out from Beaumaris, and Beaumaris Castle is impressive. Then there's Plas Nywedd (NT) on the Menai coast, a grand house and lovely grounds with red squirrels in residence. And just outside Llangefni there's the Oriel Ynys Mon - Oriel Kyffin Williams Gallery, a museum/gallery with cafe, shop etc.
Gosh, I can't wait for lockdown to ease to get back to some of these places...