Any suggestions on the best way to spend 3 days based out of split? Things to do there or by day trips from Split with no rental car?
That's a long time for Split itself. You should definitely take a bus ride to Trogir to spend an afternoon/evening there. You can also take a day trip to one of the nearby islands.
Here is a website with some good suggestions:
We spent 3 full days in Split in 2016. Because we were going to the island of Korcula after that, we didn't use any of our Split time day tripping to Hvar or the other nearby islands. Our itinerary was:
Day 1: explore Split palace/old town/harbor
Day 2: have a picnic lunch and day of swimming in the wonderful Adriatic waters at a beach just north of the monistary followed by dinner and live stage performances of traditional song and dance outdoors on the promanade between harbor and old palace
Day 3: day trip to park Krka with a small tour group; for 23€ per person, it included breakfast at an outdoor restaurant at the north end of the harbor while we waited for everyone to show up and the air-conditioned bus to arrive, then the drive to the park and a guided tour through the outdoor museum, followed by plenty of free time to swim in the lower lagoon before driving back to Split
Another high point of Split is the Architecture/History museum. Easy walk from the harbor. CHOCK full of amazing artifacts.
Another day excursion from Split is the ruins at Salona, a short trip away. Not sure about transport.
As to Trogir, you could bus there, or you could take a boat ride. We did the boat - lots of fun, great views of the shore.
Hike up Marjan Hill for a great view of the town. Take a ferry one day to one of the islands (Hvar, etc.). Tour Diocletian Palace and walk around the town. If you get bored, take a short bus trip to Trogir. See Trip Advisor for more ideas. I really liked Split!