http://www.gohawaii.com/big-island/guidebook/first-trip

I suggest that you arrive at Kona Airport (cheaper than using Hilo Airport), get a rental car, and travel south... If you plan to stay over night, find a hotel in Hilo.

From Kona, you go south to visit these sites on Day 1.

Historic Kailua Village (Kailua-Kona), Hawaii Island

http://www.gohawaii.com/big-island/regions-neighborhoods/kona/historic-kailua-village-kailua-kona

Ka Lae (South Point), Hawaii Island

http://www.gohawaii.com/big-island/regions-neighborhoods/kau/ka-lae-south-point

Punaluu Black Sand Beach, Hawaii Island

http://www.gohawaii.com/big-island/regions-neighborhoods/kau/punaluu-black-sand-beach

Hawaii Volcanoes National Park, Hawaii Island
http://www.gohawaii.com/big-island/regions-neighborhoods/kau/hawaii-volcanoes-national-park

Big Island map

Wind-blown tree in Ka Lae

On Day 2, you use Rt. 200 to go west and north... Make sure you get back at Kona Airport in time to return your rental car and catch your flight.