You should fly into the Kailua-Kona Airport (KOA). You can also fly into Hilo, but be prepared for a 2+ hour drive to the other side of the island. If you sign up for flight notifications at www.google.com/flights you will be notified when prices drop. Flights on Southwest are not included, so check there separately too. We typically find the best prices a few months out.
The Big Island is our favorite because of all of the places to adventure! We highly recommend getting a group together and sharing a rental car. We find the cheapest prices at Discount Hawaii Car Rental, they also give you add on drivers for free. The earlier you book the cheaper.
There are a few taxis available at the airport and pick ups elsewhere can be arranged by appointment.
Uber and Lyft are new to the island, but there are a few of them. They are better for quick trips to dinner or the airport but you won’t be able to find one to pick you up at a random hidden beach or a faraway resort.