The historic centre is small enough to explore on foot — most of the main sights are within a few blocks of Plaza Independencia. For longer trips, taxis and ride-hail apps are cheap and plentiful. The city also has an extensive local bus network (called colectivos) that runs on a rechargeable card system; you can buy and top up cards at kiosks, but for short visits a taxi is usually simpler.
Drivers in Tucumán are aggressive by Argentine standards and traffic can be heavy in the centre during the day. Walking is generally pleasant in the centre, though the summer heat (December to February) can be punishing — plan sightseeing for the morning or late afternoon.