Split is a fabulous city for mooching around and discovering places, I’m envious! I have quite a lot of family there.
The best restaurants and cafes can be found by getting lost in the tiny alleys of the old town - just enjoy wandering around and see what takes your fancy. My great grandmother used to have an apartment on the riva, now all smart looking restaurants and bars but while it’s nice for a stroll, I’d recommend exploring and getting a little lost.
A gentle stroll up Marjan at one end of the riva will give you gorgeous views across the bay and the city.
If you wanted a beach you can either go to Bacvice (batch-vee-tzeh) which is walkable really from the centre and the sandiest (it’s mostly rocky or just concrete jetties dotted around elsewhere). Or alternatively get a bus out the other direction from the riva through the woods/nature park past Marjan. If I recall, go as far as the bus goes before turning around and there’s a nice rocky beach with shady trees etc there. Whichever beach - if you’re going for a few hours I would take some food rather than counting on getting stuff there, other than drinks and lollies perhaps. Bacvice also has an open air cinema at night which is a lovely way to spend an evening for a change (Ljetno Kino) but not sure if it’s just in high summer.
There can sometimes be performances around Diocletian’s Palace - I haven’t seen any but my mum raved about them before.
There are fabulous things to do as trips from Split too but if it’s just a few days you might have plenty to do in the city itself.
I know you’re not hiring a car which is sensible in Split as the restrictions are incredibly strict and cars easily get towed away, but do try to stay in or near the old town so everything is walkable as the buses aren’t fantastic.
There are brilliant Airbnbs if you haven’t arranged accommodation yet, but also great hotels too of course.