Amarillo is a car city. The distances between attractions are long and the city sprawls across the flat plains in every direction. A rental car is essentially required for any serious exploration, including reaching Cadillac Ranch (on I-40 west of downtown) or Palo Duro Canyon (30 miles south).
The city has a bus system, Amarillo City Transit, but it runs infrequently and does not cover most tourist destinations. Rideshare services are available in the central areas but can be unreliable for trips to the outskirts. If you are arriving by car on a cross-country drive and sticking to the main I-40 corridor — the Big Texan, Cadillac Ranch, and downtown — most stops are within a short drive of the highway.
The historic 6th Street district is compact and walkable once you are there, but you will need a car to get to it from most hotels.