Main part of thinks which you should see is concentrated around the palace, so walking should be ok, if you want to see whole Split, you can take bus, the metro bus system works very well here.
Nearest beach is Bacvice, which is around 1 km from the old town.
You can browse the page of find-croatia for more information, or have a look at my blog where you can find a top 20 guide to split
Regards
Morten