Bozeman Yellowstone International Airport: This is the sensible airport: Livingston is about 35–45 minutes east by rental car via I-90. Rent a car at BZN if Yellowstone, Paradise Valley, fishing access or hikes are part of the trip; expect roughly US$60–150 a day before fuel and insurance, depending on season. A taxi/rideshare can work for a town-only stay, but availability is thin and an airport-to-Livingston ride is commonly around US$80–130 each way; do not count on easily finding a return car late at night.
Billings Logan International Airport: A workable fallback only when flights or rental-car prices are materially better. It is roughly a two-hour drive west on I-90, so rent a car; there is no useful scheduled public-transport transfer to Livingston. A hired one-way transfer is expensive enough that it rarely beats renting.
Livingston Depot: The handsome 1902 depot is a museum, not a passenger-rail station. Livingston has no Amtrak or practical intercity rail arrival, so do not plan an itinerary around arriving by train.