You can undertake excursions/daytrips and explore the Sierra Maestra, go to the beach or Parque Bacanao from Santiago.
I recommend a trip to 'El Cobre' a pilgrim's church just outside of town (by taxi), well worth seeing and for a Christian a very impressive experience.
As for beaches ... there are many around with different characteristics ... though none have white sand and/or a great deal of infrastructure. Try 'Siboney'- the beach most frequented by tourists. Expect some hussling, or 'Mar Verde' or 'El Frances' all about half an hour's taxi ride from town.
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