The historic core — the bazaar, the mosques, the Arg, the citrus gardens — is compact and best walked. From the Vakil Bazaar to the Pink Mosque is five minutes on foot, and from there to Shah Cheragh another five.
For the tombs of Hafez and Saadi, the gardens out at Bagh-e Eram, and the Quran Gate, you will want a taxi or the Snapp ride-hailing app, which works well in Shiraz and is much cheaper than negotiating with street drivers. A typical cross-city ride is under a dollar.
The metro has one line that runs north–south, useful for reaching the northern suburbs and the Tomb of Hafez but not much help for the central sights, which lie east of the line.
City buses are slow and confusing for non-Persian speakers. Stick with walking and Snapp.