My wife and I are thinking of taking a ten day trip to the big island in late June. Any advice of where to stay and what not to miss would be appreciated greatly.

If you want sunshine without it being too hot then the resorts on the West side of the island are the way to go.  Kona has hotter temps and a much greater selection of bars and restaurants but not much for beaches.  Hilo is rainy and I wouldn't recommend it.

Volcano National Park, Waipio Valley, and the top of Mauna Kea are good places to visit.

If staying at a resort isn't a priority for you, I'd recommend renting a condo in Waikoloa.  The west side of the Big Island is much better for tourists and has a lot more options than the Hilo side.

I second what the previous member said. Kona and the whole island is spread out and mass transit is not as plentiful so a rental car is a must.

Check out several coffee farms and drink at the local coffee houses. you can get starbucks anytime.  kona beer brewing company has some good food