Go to Alum Bay, but don't stop at all the usual tourist shyte, either walk to the Old Battery, or take the double decker bus along the cliff edge and then have a wander around the old battery, then make sure you walk up away from it and have a wander around the gun emplacements and then further south to see the Black Knight rocket launching site.
Natural Trust do tours of the fort on top of Culver Down at the other end of the Island:
http://www.subterraneanhistory.co.uk/2010/08/culver-down-battery-bembridge-fort-isle.htmlSee if there's a tour on when you're here and book using the contact details here:
http://www.nationaltrust.org.uk/things-to-see-and-do/view-page/item351851/271279/If the weather's good, bring a bike and ride the Tennyson Trail from Newport to Freshwater - stunning views and bloody good exercise.
Go fossil hunting on Compton Beach.
Park up at the Blackgang Chine viewpoint car park and walk up to the Pepperpot (old lighthouse) at the top of that down. Stunning building and great views.
Explore the Undercliff area near Ventnor on foot - Europe's largest rotational landslip.
Feeling flush and hungry? Lunch at The Pond Cafe in Bonchurch and dinner at the Hambrough in Ventnor. There is also great tapas in El Toro Contento in Ventnor and good modern cuisine with a cracking view of the Solent at the Little Gloster in Gurnard.
Park up in Cowes and walk the Boat Trail, shown here:
http://www.gifttonature.org.uk/uploads/content_documents/552/Cowes_Boat_Trail_Leaflet.pdfWhich includes several minutes standing at the floating bridge terminals wondering how the f*ck it's possible to run a service such as this with no f*cking timetable.