I have always done direct flights but that's not always possible for Kauai. In those cases I arrange an interisland flight but I don't wait to do it on the spot. I'd rather know how long the layover will be and if I can check my luggage all the way through.

Depends where you are flying out of. Some places have direct flights to all the islands. Some you have to go thru Honolulu.

So any/all of the things you list are possible and the way some people do it.

Suggest you google away and find the best deals. From the main/International airport in Honolulu/Oahu - if you haven't checked your baggage through - you can just grab your luggage and head next door to the Inter-island terminal, a short walk of take one if the wiki-wiki's, whatever they are called.

The flight to Kauai takes about 20 minutes - (Big Island - 40? minutes) - and once we land - I usually head across the street to the rental car counters - while my wife waits for the luggage, or vice versa.

Thanks all, you confirmed my suspicions I need to spend more time with Kayak and figure out how to maximize my time. I figure I cannot risk booking separate in case my first flight is delayed, etc.

Like suze says, depends where you are flying from, I fly out of LAX and there are nonstops to Kauai on AA, United and sometimes Delta. I have also flown Hawaiian with change in HNL, if you do that, be sure and book all on one ticket, as due to a recent HA policy change, if you book on separate tickets you have to claim your baggage in HNL and recheck it for the inter-island flight, might be a new fee as well. Allow at least two hours between flights, go have a drink at Stinger Ray's.

If you plan on visiting the Big Island you might consider flying into Hilo and out of Kona, or vice versa. For the transition day(s) from one side of the island to the next
do try to spend some time at Volcanoes National Park
or other stops that might interest you.

You asked:
"...or is it possible to get tickets to another island from Honolulu on the spot... "
It's my experience that inter-island flights fill up fast, so you might want to consider planning this.

R5

If you're coming from the UK, there are several mainland US cities you can use to connect directly to Kauai or Big Island without going via HNL. Some of these are on 757's; I'd hope for a widebody craft.