For your other island, here's the thoughts from a local (warning: I've only lived on 2 of the islands, Oahu and Maui, but have relatives on 2 other major ones: Kauai and Big Island. Have yet to step foot on Lanai or Molokai so won't comment on those.)
Maui is a happy medium between industrialized/built up Oahu and rural/provincial Big Island. We love visiting my cousins on the Big Island for New Year's since they have the best fireworks, but on the Big Island there's a lot lot lot of LAVA fields. And more lava fields. However, the big selling points there are A) Kilauea, an active volcano, which sometimes erupts and you can see magma coming out of the earth (I've only seen it oozing, but its pretty awesome - they let you withing several yards of certain areas many times). Probably one of the more 'unspoiled' and underdeveloped islands. Its also BIG, with 5 volcanoes which means lotsa driving if you like that. The islands look small but when you're driving around volcanoes it takes awhile since you can't go from point A to B easily. Also slightly cheaper than Maui.
Kauai is also pretty rural, and less developed, has the world's wettest spot with cool rainforests. You'll probably get a better feel for "Old Hawaii" here, but not as many modern conveniences (ie. no Costco or
Walmart or really malls so if shopping is a priority, not so much here, also kinda bad if you're hoping to cook a lot and don't want to spend an arm and a leg on groceries), not as many restaurant choices, etc.
All the major islands offer wonderful beaches, helicopter rides, etc, etc. And the part of the island you choose to visit makes a difference (Hana vs Lahaina vs Kaanapali vs Wailea vs Kihei all very different vacations, some more accessible to each other than others).
So it really depends on taste - you'll have some fabulous restaurant choices about a half hour drive from Aulani in Honolulu that will get you a good feel for the island that is developed, and for me I really like civilization. But if you're looking for a more undeveloped area with a small-town feel, then Big Island or Kauai. If you were going to one island, definitely Maui because its the best of both worlds, but since you're already going to see the city on Oahu, perhaps Kauai might be a good second choice since its the opposite of Oahu. Unless you don't plan on leaving the Disney resort on Oahu, in that case come to Maui. Or want to attempt to see Kilauea. More developed = more to do, but also loses some small town charm.
Sorry if I confused you more. PM me if you have any specific questions I can help with.