Just about everything you want to see in Split is within the walls of Roman emperor Diocletian's Palace, completed in 305 AD. It's a compact area and you'll easily find your way around the narrow alleys. Make sure you visit St. Domnius Cathedral, well over 1,000 years old. You could climb to the top of the bell tower there for a view of the city, but I found a better spot: Walk exactly 10 minutes to the left of the palace and then up some (well, more than just "some") stairs and paths in the neighborhood of Marjan. By the time you reach the tiny St. Nicholas Church (1219), you'll be rewarded with an even better view, without any tourists at all (I was there alone, with a baby green lizard passing by), and it's free.
In town, the outdoor markets sell everything from prosciutto to pantyhouse. The waterfront promenade is delightful, especially at night, when it's cool and the locals gather, and they all seem to know each other. I was fortunate enough to stumble upon a band performing there, so keep your eyes and ears open as you stroll along with your refreshing ice cream cone.