I agree with PP that you will most definitely want to rent a car. You did say you're DVC and I assume you're staying on points, so parking will be free. Renting a car gives you the flexibility to get to and from the airport, make grocery stops, visit wherever you want, and easily be able to go off site for meals.
We did our grocery shopping at Target in Kapolei, which is the next stop on the highway, and found prices to be mostly comparable to the ones back home in NY. With a family of 5, you may do better at Costco if you're members, which is also in Kapolei. There's several chain restaurants and reasonably priced restaurants in Kapolei that have prices more in line with what you may be accustomed to back home. Keep in mind that Ko Olina, where Aulani is located, is a planned gated tourist community. Everything at Aulani and surrounding resorts, as well as the few restaurants and shops in walking distance of the resort, will have tourist pricing which is as high as you'd expect.
There's a lot you can enjoy on Oahu with little to no cost. Aside from what PP mentioned, the Haleiwa area is beautiful and fun to stroll and stop at the different beaches along the way. Laniakea Beach is a fun stop to see turtles. We also loved strolling along and hanging out in Waikiki - fun atmosphere and beach. I would avoid doing Waimea Valley which has a hefty price tag of $16 per person, which we didn't find worth it. We also loved the Kailua area and the beaches there were the best we visited. I'd also suggest Kualoa Ranch and I believe they have tour packages which offer savings as well. If you're interested in a luau, the Paradise Cove luau is walking distance from Aulani and I believe is cheaper than the one at Aulani.
We didn't use it, but there's also a Go Oahu pass/card that includes multiple attractions, depending on what you're interested in. Aulani offers some excursions, but with a car, you can likely book it yourself at a cheaper price point.