From: Federal Way, WA
The Big Island doesn't have a lot of attractions compared to the others, and what is interesting to one will be completely boring to another. That said, here is my list:
1. Visit South Point (but avoid any scuzzy-looking types while you're there). You can take turns being the southernmost person in the United States.
2. Based on the assumption that you're staying in the Kailua-Kona area (since you're there for a week), drive the Queen Kaahumanu Highway (Route 19) around the northwest tip of the island (at the junction with Route 270, turn left onto 270). Just past Hikapoloa take the Upolu Airport Road to the left. The airport itself is interesting to anyone with an aviation bent (5 whole planes based there, winds that make me wonder how anyone survives flying there), but it's just an interesting drive. After backtracking to the main highway, continue in the original direction. In the town of Hawi, turn right on Route 250 (it's been a while, but I think it's the only stop sign in town). This takes you back to Route 19 in Waiaka. From there you can turn right to head back to Kailua-Kona, or turn left and head into Waimea. Route 250 is NOT the type of road you expect to find in Hawaii; the scenery is more reminiscent of California! If you go to Waimea, either turn right at the junction with Route 190 and you can head back to Kailua-Kona (the Kamuela airport, just outside of town, is another throwback diversion for airport junkies) or turn left, staying on Route 19, and head for Hilo.
3. On the road to Hilo, there is a railroad museum in Laupahoehoe. It's not big -- you can see it all in under an hour -- but has a good display of island history in addition to the trains. They're happy to get any visitors that stop! Beware; the woman who works there can talk your ears off!
Back on Oahu there are, of course, zillions of things to do. Just a short list of my favorites that aren't on everybody's list:
A. At the east end of Waikiki (to the left as you face the water) there is an old shore battery that is now a museum. Modern stuff on the roof, but the inside (nice and cool, by the way) is all World War II. They can't tear it down because the necessary blasting would level half of Honolulu, so they're making good use of it.
B. Near that battery is the Honolulu Aquarium. One of the best I've seen (but still not the Tennessee Aquarium, in Chattanooga). You definitely see a lot of fish you won't find in most others in the US.
C. Of course, there's a railroad museum on the island! In Ewa, just west of Pearl Harbor, the museum has a lot of railroad cars that were operated on the islands while the railroads were still there (and they were more than just sugar cane haulers -- Oahu had an extensive commuter rail system at one time). On Sundays there are two trips on the train out to the west side of the island, stopping near the Sheraton resort out there (they don't have a choice; a few yards farther on the tracks go straight down). Some interesting scenery along the way, and while the trip is narrated it's not the overly-touristy "Sugar Cane Train" experience on Maui.
D. I've never been, but the Pacific Lightning Museum is at Schofeld Barracks.
E. Hale'iwa is an artist colony on the north shore, but of interest to a World War II buff for its role as an auxiliary airfield. Dillingham airport to the west is still there, too, and has a major glider operation.
F. While designed and priced for tourists, the Dole Plantation is still an interesting stop (it's on the road up the center of the island, past Schofeld Barracks but short of Hale'iwa). Catch one of the demonstrations of how to cut a pineapple; it's a lot easier than most mainlanders think!
Sorry this is so long, but it's one of my favorite states to visit. I could go on and on!