The old city, fort and main bazaar are walkable, though the streets are narrow and shared with carts, scooters and the occasional cow. Auto-rickshaws are the standard way of getting between the railway station, the fort and the Mall Road area — agree the fare before you set off, or insist on the meter.
Cycle rickshaws still ply the inner lanes and are perfectly fine for short hops. App-based cabs (Ola, Uber) work but cars are limited; expect waits. For day trips to Damdama Sahib or Maiser Khana, hire a taxi for the day from the railway station rank.