Welcome to the thread! You'll be amazed at how much information you can find on here about so many things. I'll give you my input on your idea.
First of all, like everyone said, be sure you do a lot of research on what will be the right boat for you and your family. My beau and I did a ton of research on the perfect boat for us when we decided to move onto a sailboat. We wanted something in the 33-36' range, and some of the 36' boats felt a lot more cramped than some 30' boats we looked at. So find something that is going to suit your needs and allow you and your family some breathing room.
However, don't just get a boat that feels big without doing the research on safety factors. Capsize ratios, motion comfort, etc. need to be taken into consideration. ESPECIALLY if you plan to cross an ocean. While people have done it with little experience, I would not recommend taking a course to just get the basics, "enough for a 30 day voyage", as you mention. A 30 day voyage is a long time, and so many things can come into play, that unless you are an experienced sailor, I would recommend finding a captain or someone to assist you with the journey. I absolutely love going offshore, but it isn't as glamorous as people sometimes think. Try going 7 days with no shower!
I'm in no way trying to discourage you from doing this - just be sure that you know what you're getting into before you embark on a journey that big.
As far as staying on a mooring in Hawaii, it's doable, but like several have mentioned, just be aware of the realities. 4 people on the boat using the head - you'll need to be near a pumpout station where you can pump out regularly. Then there's showering, electricity, etc. A marina sounds like the better option for you, but they can be expensive. I'm not sure about the rates in HI, but I would imagine they are not cheap. Our marina fees that we are paying now in the Chesapeake are almost as much as the mortgage on our little house we sold to get here. Of course, we don't mind because we have some of the most awesome scenery, so it's a tradeoff! Much better than seeing our neighbor's unkempt yard every day!
Maybe you could start out by finding a boat and keeping her there in CA for a couple of years first. Try it out, get it ready for a longer voyage, learn to sail, etc. That way, if you absolutely love it, you'll be ready for it, and if you don't love it, then you can change your mind and move back ashore while you're still in familiar waters.
I wouldn't take back our decision to move aboard for anything. I wasn't sure how I would like it, and now I wonder why it took us so long to do it. So if you have the right frame of mind, and you aren't tied to a lot of material things, you can be very happy, free, and satisfied!
You can read about my personal experience with the decision to move aboard here:
Does Size Matter? | Vacilando
Best of luck to you and your family! Can't wait to hear how your search comes along!
Vacilando | This is the story of two people, a dog and a journey. We'll choose some article almost certain not to exist and diligently try to find it.