The Tasmanian Museum and Art Gallery across the road from the wharf is my favourite museum. It is free with donation tag.tas.gov.au It has a courtyard cafe
A short walk along the wharf is Salamanca, a delightful place to explore with all its small unique shops and cafes. There is a big market every Saturday from 8.30 am to 3 pm. www.salamancamarket.com.au
The beautiful Royal Tasmanian Botanical Garden is well worth a visit (local bus or taxi).
The CBD with its shops is walking distance from the wharf. The Farm Gate Market in Bathurst Street is on every Sunday from 8.30am Great fresh food and farm products.
Battery Point 10 mins. walk from Hobart CBD. "Climb the colonial era Kelly's steps and wander up through the lanes and quaint streets of Battery Point filled with stories of Hobart's historic past." tasmania.com
There is so much to see and do around Hobart's Wharf, you don't need a car on the second day.