Regarding Hobart, as CPMaverick said most places will be closed on Christmas Day so plan/book your meals ahead. But yes a lot of the touristy places and restaurants will re-open on Boxing Day. If you hire a car, you can see much more outside of the city, which I would recommend for both foodies and nature lovers. Bonorong will be open on Christmas Day 2018 so you could combine a morning there with Mt Field (short walks to waterfalls, tall forests) or even do the Strathgordon / Gordon Dam road if you are happy driving for ~5 hours return.
You can use the Mona ferry or Mona bus to get to the city centre and waterfront, or you can use the public bus (metrotas; X20, 520, 522 Bridgewater bus; ~35 mins; 2 zones fare). The bus runs later (though only until 9:10 pm on public holidays, and that probably excludes Christmas Day) and costs less, particularly if you buy a Greencard which has a daily cap. But at night you'll probably have to call a cab.