We also lived for 3 years on Oahu (DH was in the Marines). Moved from there to Maryland in 1998, but have been back several times (and now have a timeshare at Ko Olina on Oahu, so will go every other year now, and did in 2008 and 2009).
I recommend Oahu for at least a few days, if it's your first trip. There is SO MUCH in my opinion on Oahu to see and do.
To the other poster: the luau that picks you up is Paradise Cove. It is on West Oahu, right in the same resort area as Marriott's Ko Olina (timeshare, AWSOME), the JW Marriott, and where Disney is building their new DVC resort. I've done it twice, once in 1995 and also ten years later. I don't know, the show gets awfully long, it is cute, but I've no desire to go to another luau unless it's a friend's party or something.
Snorkeling on Oahu: DEFINITELY at Ko Olina is really excellent. They have a few man made lagoons that are very calm and not deep (except in certain areas). My DD snorkeled every single morning last year, and we saw all sorts of fish and turtles. It was very cool! On weekends and holidays, it's a favorite spot for locals to hang out with their families and picnic, as it's a nice clean area, safe water, etc. My kids met some other local Hawaiian kids and played together in the water one day.
What I liked about staying at Ko Olina (we used our week plus paid for 4 extra nights last year) was that
1) we had a full kitchen and washer/dryer (we did laundry every day as we'd be all sandy and such)
2) they have gas grills that they clean daily for you to cook with, and they are very popular at Ko Olina
3) the new pool with rock slide and waterfalls was a big hit with the kids
4) there is a very nice Costco and a new Super Target about ten minutes away for groceries, and Costco had excellent fresh fish to grill as well as sashimi and yummy pineapples, so we cooked at least two meals a day in the condo which was not only economical, but the kids were more relaxed than going out to too many restaurants, and I didn't mind as I made easy stuff and grilled several times
5) it has the neighbor island feel, but you are 45 minutes from Honolulu, so we did walk Waikiki one day and also ate at Chai's Island Bistro at Aloha Tower one night (to see Nathan Aweau of Hapa)
For Hawaii on a budget, I would definitely recommend finding a condo or rental home. You will need to do laundry! Pack light, but I agree bring your snorkel gear (we got ours at Costco, and I saw some at the Costco in Kapolei). If on Oahu, pick up souvenirs and tee shirts ($3 each) at the Aloha Flea Market at the Aloha Stadium on Wed, Sat and Sun's.
My first choice would be the Marriott Ko Olina, as you are at a true resort with the super pool and all. If that is too pricey, there are also rental townhomes right in the same resort area (many folks like at Ko Olina, but some homes are rented out weekly or even monthly).
Anywhere but Waikiki, you will need a rental car. No, a convertible is NOT worth it (my silly DH got one our first trip there, how stupid, first of all you LOOK like a tourist, second, you may drive through a five minute rain shower, too much up and down for the roof). Just get a cheap rental - parking spaces are small anyway!
If you love sushi (my DS does) don't miss going to any local Genki Sushi. It's the best. And super reasonable.
Have fun planning!