There is one big nursery KNL (Kauai Nursery and Landscaping). There are several small nurseries, but nothing really big for fruit.
http://www.kauainursery.com/ their fruit book can give you a starting idea of what they have, but they have much more than listed and sometimes you have to ask. Leeland is the owner and is a collector.
http://www.kauainursery.com/images/Fruitbook.pdfVisit NTBG (National Tropical Botanical Garden) for lots of fruit trees. There is a garden walk at McBryde Garden, that has maybe 50+ (mostly mango, breadfruit, lychee, banana, canistel, etc) different fruit trees as well as a history of Hawaii. Allerton garden is pretty nice to see, again mostly ornamental.
http://ntbg.org/tours/kauai-south/ This is a huge place, several different tours.
Na Aina Kai is a pretty place. They have a fun playground for kids (kids (<9 yrs old) can easily spend a few hours here. Check the hours on their website. They have a larger garden that is pretty, not a lot in the way of fruit.
Princeville Botanical Garden is also a pretty walk. Mostly ornamental, very little in the way for fruit.
There is a chocolate tour at Banana Joes that
maydo fruit tours. Banana Joes has lots of rare tropicals (probably the most rare in a commercial setting). I have not been on the chocolate tour at his place. Next to banana joes is a mini golf course that is completely gardened with plants. There are tons of things in there, some that I hadn't seen before. Its fun to go through and spot things.
http://www.gardenislandchocolate.com/book-a-tourThere is another chocolate tour at Steelgrass farms. This chocolate tour is a great introduction to tropical fruit as it lets you sample maybe 10-20 things in season. My big gripe is there isn't that much fruit or chocolate for the price (compared to other tours I have tried).
There is a farmers market almost every day of the week somewhere on the island. Search sunshine markets. There are several others that don't fall under the sunshine market's class. I would visit a couple, because many farmers only show up to 1 or maybe 2. Some of the bigger farmers attend many.
https://www.tastingkauai.com/farmers-markets/