If you're looking to escape the chaos of Split I suggest a trip to Marjan. Marjan is a large nature preserve laden with jogging trails and scenic lookouts over the Adriatic. I recommend taking the trails all the way to top, to the southernmost lookout where you'll be able to see the nearby islands of Brac and Solta.
Keep an eye open for the blue agave plant which is famously used to make lace on the nearby Croatian island of Hvar. I was surprised to see this plant growing in Croatia but it is plentiful throughout the Marjan grounds. Just follow the Riva west and mind signage for Marjan.