Three incredible hikes to do in Kauai Hawaii. None involve a permit, all are public and free. The first trail is the Moalepe \ Kuilau trail (easy). It's a hike through a forest with an arboretum, and jungle and has spectacular mountain and valley views.
Full trail video - https://www.youtube.com/watch?v=CF7AZA5PRJo&t=48s
The second trail is the mahaulepu heritage trail (moderate). It follows the south shore along cliffs and beaches and is perfect for whale watching in the winter.
Full beach hike - https://www.youtube.com/watch?v=Sb5hdE6Bkv0&t=968s
Full Cliffs hike - https://www.youtube.com/watch?v=rJy77U_yoOo
Full cave video - https://www.youtube.com/watch?v=QA1mQ9APfCo&t=81s
The third trail is the makaleha falls trail hard). This trail is more of a jungle crawl, that passes through a bamboo forest, criss crosses kapaa stream, has an emerald green plunge pool and finishes at a waterfall to swim under. All are amazing hikes and the first two are family friendly. The last trail is a true Hawaiian adventure, suitable only for the more hardcore hikers and adventurers.
plunge pool - https://www.youtube.com/watch?v=vogJR31wXmQ
Full trail video - https://www.youtube.com/watch?v=dBSN8vAcSww&t=5s
The first trail is near Poipu on the south shore. The last two are on the East side of Kauai near Kapa'a, Wailua and Lihue. For an easy walk, the first hiking trail is best. For a more moderate but still easy hike the second trail is perfect ,with an added bonus of being in a perfect whale watching area. The last hike is a monster sure to satisfy anyone looking for a challenge with an added bonus of having wild fruit to pick along the way.