A beautiful old city! Others here have already done a good job explaining what is best to see in the town so I will keep this review fairly brief as not much has not changed since then to here in Jan 2025. My wife and I spent a day here with my cousin, who had come to Salamanca a few days before us to spend time with her graduate school friend who is a native of the city. The friend who had grown up there (a "Salamantina") somewhat cheekily put the things to see in Salamanca into three categories:
Old & Beautiful
Beautiful, but not Old
Old, but not Beautiful
Sites like the Catedral Nueva and Catedral Vieja (which are one combined entrance fee & location where you pay to see both with an audio guide) fall into the coveted "Old & Beautfiul" category. Other things like the Roman Bridge are in the "Old, but not Beautiful" classification.
We were kind of just wandering the city without a specific agenda or checklist to complete, so a hitlist of items that we ended up seeing that were definitely worth it include:
The Cielo de Salamanca in the ancient Salamanca University - An ancient astrological painting inside Spain's oldest university
Convento de San Esteban - A ridiculously intricate alterpiece that is dozens of meters tall
Plaza Mayor - The typical "main square" of older Spanish cities, and very similar to the Plaza Mayor in Madrid
Definitely a worthwhile Spanish World Heritage Site to visit, and it makes for a very pleasant daytrip!