There are two main airports in close proximity to Yellowstone National Park and several smaller fields for commuter flights and private planes. The one north of the park is The Bozeman Yellowstone Airport at Gallatin Field. This arrives into the town of Bozeman where you may find fine accomodations or rent a car for travelling to accomodations closer to the Park.
The airport south of Yellowstone is Jackson Hole Airport. From here you can rent a car and Travel through Teton Park up through the Rockefeller parkway into Yellowstone National Park.
Once you have arrived in the area it is best to rent a car for the rest of your travels. There are no real shuttle services which will take you to the trailheads in the Park, and hiring a cab is not cost effective. Trailheads are fairly secure and your vehicle will be parked next to ours at the trailheads. Rangers patrol these areas fairly regularly.