Everyone who goes to Kauai sees the major sights such as Waimea Canyon, Fern Grotto, Poi’pu Beach, etc. However, many other stops in the Garden Island are worth spending time on.
The best spot that few tourists see is the Kauai Museum. We stopped because the weather was iffy and ending up spending several hours at the museum. We spent over ½ hour in the "theater" watching a beautiful video of Kauai. Some of the footage (for instance of the beautiful NaPali Coast) was taken from a helicopter. Watching the waterfalls and other scenery was not as great as being on a helicopter myself, but it came close. And a $5 adult admission fee to the museum is a lot cheaper than $75 (or more) for the real helicopter ride. The narration of the video was very professional and informative. After watching the video, we visited the rest of the museum and enjoyed the exhibits. Anyone who loves quilts would love the large display of Americana quilts. Many photographs depict the history of the island.
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Another spot we visited that was totally deserted by tourists was Hanalei Bay. Many tourists stop at the Hanalei Center and Princeville but our walk along the Hanalei Beach was great. The black sand is very fine, the water is calm, and the entire area is so peaceful. While there, many local artists were setting up easels to paint the scenery. People also fish along the long pier. This beach is off the beaten path but just as nice as some of the more popular (and more crowded) beaches.
I would also suggest that you take the time to visit a "crafts" event. While we were in Kauai, we stopped at the Kauai Products Fair. Most of the vendor booths were selling art, jewelry, or food. The art was very unusual and very good. I was very tempted to buy some articles. My wife was absolutely fascinated by the jewelry and she did buy several pieces there. Each was handcrafted. Click here for more information.
One suggestion is to pick up the 101 things to do pamphlet and try to sample some of the lesser-known "attractions."