When you say “the big island, honolulu” you are aware that those are two separate islands–The Big Island (which has two major cities: Kona & Hilo), and Honolulu is the capital city on the island of Oahu? Just checking because your post made it sound like you were thinking of them as one place.
They are quite different, but special in their own right. Oahu is much more urban and Honolulu is a bona fide city. The Big Island (TBI) is quieter. So it depends on what kind of experience you want (although you can easily do both because the flights between island are like, 45 minutes). TBI has the volcano and a national park surrounding it, so you can do fun things like hike in the rainforests and see the lava (albeit from a distance!) and fly over the flows in a helicopter. Because of the volcanic activity, the topgraphy of TBI is oftentimes black and sort of desolate looking (I think it’s beautiful, but others find it a bit too close to the landscape of Mars in black), and they have famous black-sand beaches. Kona is the dry side of the island; Hilo is the wet. There are some coffee companies out there that you can tour and the old Hawaiian village with the turtle sanctuary and petroglyphs, but on the whole, TBI is rural and quiet so there isn’t much by way of entertainment outside of the resorts.
Honolulu is a city. I repeat: Honolulu is a city. That doesn’t mean that you can’t find some form of paradise on Oahu, but sometimes people go expecting their desolate white-sand beaches and quiet paradise and they’re shocked that Honolulu has terrible traffic and lots of tourists. Don’t get me wrong–my family is from Honolulu and it happens to be my favorite island for many reasons. Just understand what the experience will be like. The trade-off is that just like visiting any other city, there’s tons to do and lots to see: great beaches (our family still uses Waikiki, tourists and all, because you can surf there as a non-local, it’s close to where we live, and parts of it are safe for little kids, but if you drive, you can find some less-well-trodden places), great hiking (Diamond Head and above Lanikai/Kailua to name a couple), museums and art galleries, historical sites like Pearl Harbor, shopping, restaurants, nightlife… But yeah, expect something like Rio–a city on the beach–rather than like, a faraway hut in the Cayman Islands or something. If you want more of a honeymoon experience on Oahu, I would suggest that you look beyond Waikiki. I can’t speak for hotels, but I know others who have rented private homes instead and found that to be more private and intimate. If it’s any interest to you, Lanikai, which has one of the best beaches and is quite tony, is the windward shoreline that’s north-east of Downtown Honolulu, is where Obama stays when he’s in town 🙂
Either way, if it were me, I’d definitely rent a car on both islands.
And so long as you are looking at Hawaii, you might want to check out Kauai and Maui as options too. IF you’re looking for the quintessential lie-on-the-beach-making-out honeymoon deal, then those islands might be more your speed. I only mentioned more about Oahu because most flights go into and out of Honolulu (you CAN of course, spend no time in Honolulu if you don’t want to and just puddle-jump OR find direct flights to outer islands, but they might be more expensive). My one tip to that end is IF you arrange flights that go from the mainland to Honolulu, then you puddle jump to an outer island, then give yourself time between the flights because last time I checked, in almost all cases, you will have to pick up your bags from baggage check and re-check them (this is unless you only have carry-on), plus switch terminals. There’s exceptions to this rule, but on the whole, make sure you know what the deal is because Hawaii is very careful about agricultural checks on and off the islands AND between islands and it’s more often than not that you’ll have to collect bags and re-check. Just a heads up.