I would never recommend driving on Oahu to the unseasoned driver, especially not a newly licensed one.
OP, I recommend staying somewhere other than Aulani if you want to experience Hawaii and not the Disney version of it.
Aulani is not well-situated for someone who wants to see the rest of the island.
I suggest staying somewhere in the Waikiki area to experience the touristy stuff that is all within walking distance and where you will find competitive pricing for tours and other excursions that require pickups at your hotel.
I totally agree with this.
By the way, if budget is a concern, have you considered looking into renting a 2 bedroom villa at the Marriott's Ko Olina Beach Club instead (walking distance to Aulani, so you can go over and eat/drink but of course not swim, but the pools at Aulani are WAY more crowded than the Marriott)? You can look at
www.redweek.com and compare weeks. I've rented when I needed an extra week.
Honestly there is no way I'd go for less than 10 nights though. It's just too far. I suffer little jet lag, but take melatonin (I have since we lived there 23 years ago and I'd bring it when I visited the mainland).
We use airline miles so have never paid for a ticket, even though we've been back over a dozen times. I play the points game, and used to sometimes fly coach, but I refuse to any longer. I want a lie flat seat especially for the red eye coming home. Cash for those flights was running about 4K per seat but miles worked out for our last trip and my next one. I follow the points guy and am on flyertalk.com a lot. There is a lot to learn, but you may want to look into credit cards for miles/points. My favorite over the years has been the Starwood amex.
There are many reasonably priced hotels in Waikiki as well. They don't have to break the bank, and if it's just a hotel room (not a villa), we're not in the room much anyway. Just to sleep and shower. If one doesn't want a rental car, Waikiki is a much better option.