To be honest, it depends what you want to do. If you are huge Disney fanatics, Aulani is great. I've had staycations there where you really don't need to leave the resort. There's stuff to keep you constantly busy. If you want to see the rest of the island, however, most people veer towards Waikiki because it's near town. Waikiki is the more "touristy" area so it tends to be very crowded and flooded with "touristy" trinkets which is both good and bad. Waikiki has a bunch of really good restaurants and the beach is right there.
Aulani is WAAAAAAY on the other side of the island near Kapolei. Kapolei is expanding too though so there are a lot of stores and things on that side that weren't there before. Our only water park is in the area as well. The good thing about Aulani is that they tried to keep the resort very true to the culture and not "touristy." We even have cultural advisers to help keep it in check.
The North Shore is also a cool place to explore but it's not close to Waikiki OR Aulani...
Honestly though, compared to the mainland (is that a Hawaii term? lol) any drive is relatively short
You can drive around the island in about 2 hours
How long are you planning on staying on the island and are you planning on island hopping or just staying on Oahu?