Phoenix Sky Harbor International Airport: Use the free PHX Sky Train from Terminal 3 or 4 to 44th Street/Washington, then take Valley Metro Rail west to Downtown Phoenix (roughly 20–30 minutes from terminal to downtown; about $2 for a single local transit ride or roughly $4 for a day pass). It is the sensible budget transfer. A rideshare is better for Scottsdale, resort areas, or late arrivals: downtown is about 15–20 minutes and commonly around $18–30 before surge pricing.
Phoenix-Mesa Gateway Airport: This is an Allegiant-heavy airport in far southeast Mesa, not a convenient Phoenix airport. Take a rideshare or prearranged rental car; expect roughly 35–55 minutes to downtown Phoenix in normal traffic, longer at rush hour. Public transit exists but is too slow and fragmented to recommend after a flight.
Maricopa Amtrak station: Phoenix has no Amtrak station. The Sunset Limited stops at Maricopa, about 35 miles south of downtown, only three times weekly. Use the Amtrak Thruway shuttle when its schedule matches your train; otherwise take a taxi/rideshare. Budget roughly an hour door-to-door to central Phoenix, and do not assume a frequent local connection.