Lihue Airport is the entry point for Kauai and it's on the east side of the island, roughly 30 to 40 minutes from the inn depending on traffic on Kuhio Highway - the drive north from the airport to the north shore is itself a reasonable introduction to what the island looks like, the road following the coast through small towns with mountains visible inland the whole way. Rental cars are available at the airport and on Kauai a rental car isn't optional, it's the actual infrastructure of getting around, and booking early matters because availability tightens in peak season.
Inter-island flights connect Kauai to Honolulu, Maui and the Big Island through Hawaiian Airlines and Mokulele with multiple daily services - the flights are short, 20 to 40 minutes, and the logistics are straightforward enough that building a multi-island trip with Kauai as one stop is realistic and common. Arriving from the mainland usually means a connection through Honolulu, and the Kauai leg of that journey is the one where people look out the window and go quiet.
Getting around Kauai is essentially one road - Kuhio Highway runs the east and north shore, and that's the main artery. The road ends on the north shore and the south and west sides require backtracking through the interior, which sounds inconvenient and in practice means that cross-island drives take longer than the distances suggest. Traffic around Kapaa in the morning can be genuinely slow and building extra time into any schedule that involves driving south is just sense rather than overcaution.
Public transport on Kauai exists in the form of the Kauai Bus which runs a route along the coast, and it works for some limited purposes, but it doesn't serve the Na Pali trailhead, doesn't run frequently enough for flexible day planning, and doesn't handle the kind of equipment that hiking and beach days require. Most visitors figure this out quickly. Car is the answer and has been since the road was paved.