I guess I'm asking if there would be any additional issues I could encounter if I book flights separately rather than connecting. As in, if your first plane is late is it treated any differently if you are not able to make the next flight? The reason I would book separate flights is if I would want to stay a few days in another city on my way back- like fly from home to NY and NY to Hawaii then on the way back fly from Hawaii to NY and vacation in NY for 3 or 4 days before flying back home.
If you're on separate reservations for two legs of a continuous flight, you are much lower on the rebooking priority list, except for AA (and in October US) which will link your PNRs automatically - it's part of their OneWorld Alliance booking procedures to allow better interlining (multiple airlines on the same route).
All of the airlines offer multi-city booking choices, where you can pick your origin and destination and put in stopovers as you please. And as the best routes to Hawaii are on the five legacy carriers (AA [American], UA [United], DL [Delta], HA [Hawaiian], AS [Alaska]) you can expect largely similar rules between all of them. Also, it means that you only need to check prices on five individual websites, max, which isn't bad.
The multi-city ticketing is never more costly than a collection of one-way tickets, but it puts you all on one reservation for tracking and fixing problems caused by delayed/cancelled flights, so it makes sense to do it this way.
As for connecting flights, how long would you recommend leaving considering kids and bathroom breaks and possibly food between flights? Should I aim for 2 or 3 hours between or is that too long?
The minimum connection time (MCT) at most airports is 30-40 minutes. Figure an hour, if all goes right, including bathroom breaks and a quick stop for food, but as you're unfamiliar, two hours may be better. If your incoming flight is late, that two hours can get eaten up as well.
The airliners also do have lavatories on board, and do serve food, particularly on the over-ocean leg. Although airline food isn't the best, it's about the same price for similar quality food in the airport itself.
You don't say what airport you're going out of, but if you look up which airlines service it and at what levels you can get an idea of which ones are going to be best to check. If you don't have JetBlue and Southwest, you're probably in a small airport and may be limited by that alone.