Ellison Onizuka Kona International Airport (KOA) is the closest airport, about 7 miles north of town. It handles direct flights from the mainland US and inter-island connections. From the airport, take Highway 19 south toward Kona. At Mile Marker 100, turn right on Palani Road, follow it to the end, and it bears left onto Alii Drive — the heart of town.
There's no public transit worth mentioning. Rent a car at the airport; you'll need it for anything beyond Alii Drive.