Aloha JB

I've been fortunate enough to see and experience a bit of the world in the past 3 or 4 years and the last 3 weeks of my life have been the best. Hawaii is by far the best place I've been. It's the only place I've really felt at home. Youtube Israel Kamakakowi'ole & sit back and enjoy life
The islands, people, culture, history, scenery is only topped by one thing... Godiva cheesecake

Flight wise, i'd suggest breaking it up in Sydney for one night on either side. I enjoy long flights so I can catch up on tv shows but my mrs prefers to break them up. I'd imagine its probably better for the kids as well? Hawaii is just under 10hrs from Sydney and i think most flights arrive around 7am but 4 hours to sydney, a few hours at the airport and another 10 hours makes for a long day with kids and the mrs. Book exit aisle seats. I'm tall so its mandatory for me but worth it on any flight.
I'd stay somewhere like Holiday Inn on Kalakaua Avenue in Waikiki. It's good value, great place, very central to the beach, shops, restaurants. It's not right on the beach but on the other side of the road so its much cheaper. Popular with all nationalities. Places like the Moana are pretty Japanese orientated as they cater for them quiet heavily. Since its a family holiday your stupid if you don't stay in Waikiki. If kids and mrs are happy, then dad is too!
North shore is at least an hour away, you can catch buses (take up to 3 hours) or go via various operators. It's a circus at all the big name spots but how can you not soak it all in.
All the islands, bar Niihau, are worth visiting and all are different I'm told, however 10ish days isn't enough to start visiting islands properly. Your'd need weeks on them all so your not doing them justice. Next trip though right??
Shopping is pretty good i think. There's are area for the rich (Luis Vutton etc) up near the Hilton Village but most of Waikiki is surf shops and shops that suit women. ABC store will have everything you need but can be a bit hard to find

There are so many activities. The list is endless but i've spent a bit of time in places affected by WW2 (germany, concentration camps, kokoda etc) so going to Pearl Harbour was a great but sobering experience. The yank's are very proud of their service men and women and Hawaii play's such a massive part in this.
Other tours are Kualoa Ranch (great place), sea life park dolphin swim, snorkelling, various landmarks, famous beaches, the zoo is OK, cage diving, diamond head walk (dont do a tour though) circle island tours etc. Heaps to see and do. There are some dodgy blokes on the main drag offering cheap tours but it's a time share scam. Some push it, some dont.
NFI about Sept / Oct but there were waves and people surfing everywhere. Hell, even Waikiki beach has waves that are probably better than the metro area 95% of the year lol..
Heaps of restaurants, the food is generally OK but the service is usually very good. You have to tip but IMHO they want a bit too much sometimes.
There were 13 tiger shark attacks in Hawaiian waters in 2013 so you,ll be right at home

I could go on all night, but Hawaii 5-0 has just finished downloading

In short, Hawaii is fkn incredible!!
Aloha