The Big Island, real name, Hawaii, is over twice a large as all the other islands, combined. Kailua-Kona is pretty far from any where else on the island, so a car is a must. It will be a long day to Volcanoes National Park, but its a must see and do. Unless you want to spend a night on the East side, (Hilo) it is a long two hour drive each way. South Point (the most southern point in the US) will add a good hour or so to that trip. Parker Ranch and Waimea are both the other direction, and a day can be spent there too. A trip to Hilo can be made by going across the island via Saddle Road, assuming that it is in good repair, and passable without the aid of 4 wheel drive. Most of the water-falls are on the East side of the Island of Hawaii. There is a lot of lava, every where, and not much in the way of a nice sandy beach, that would be Maui, always a good reason to come back. With just 7 days for the trip, you might not have enough time for a day trip to another island, but if you do, my suggestion would be Oahu. One day there is about all one would want to do, IMO. Its just like Chicago, except it has palm trees. Pearl Harbor, the Arizona and USS Missouri would be musts. Waikiki area and the famed beach probably too. A long day trip, for sure.
Jet lag will get you for a day or two, so be prepared for that.
Google Kona and the Island of Hawaii, and you will find a lot of neat stuff to do. Another good site to look at is
www.beatofhawaii.com One good restaurant review currently on it for Kona.
Maybe next year, we'll see you on the beach on Maui. (Noi Ka Oi)
Have a good time, and congratulations.