Within Ahvaz, taxis are cheap and plentiful. Shared taxis (savari) run along the main routes and are even cheaper. The city is flat and distances between the bridges and main sights are walkable in the cooler months, though in summer (when temperatures routinely exceed 45°C) you will want a vehicle for anything beyond a short stroll.
For visiting the surrounding sites, you will need to arrange transport. Car rental is available at the airport and through agencies in the city. Hiring a taxi for the day — negotiating a price to cover Shush, Haft Tappeh, and Chogha Zanbil — is common and practical; a driver familiar with the sites will save considerable time navigating country roads. The distance from Ahvaz to Chogha Zanbil is about 80 km; a full day trip combining two or three sites requires an early start.