Well, there's the Diocletian's Palace of course - 1700 years old. It's the actual city center, so you cannot miss it . Inside you can find st.Duje cathedral, temple of Jupiter, roman forum "Peristil", the cellars under the palace...Also in the city center is the Grgur Ninski statue (at the north gate of Diocletian's palace), another city wall (from 16th century) together with Gripe castle.
South wall of the Palace is looking over the Riva (seaside). There's lots of other things in the center...There are half a dozed museums, usually free or quite low-cost
Then there's Marjan hill, a nice pine tree forest with a great view over the city.
If you like sports, you can check out the beautiful Poljud stadium.
If you like beaches, there are a couple sand ones just east of harbour, and some really nice ones under the southern slopes of Marjan.
If you're not planning to visit the islands, maybe you should think about that - Brač is closest, Hvar is fashionable, Vis is "alternative", Lastovo wild...
If you like history, you can check out Salona ruins (really nice, but in poor conditions) and Klis castle, as well as old towns of Trogir and Omiš (30 km west and east, respectively).

Hope that's enough ))

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Spaceart: Captivating * Insomnia
Photo: Hell * Fruit