We liked the Battery Point area.

http://www.batterypoint.net/

Stayed at Lenna's of Hobart. It was okay, old (you might say in need of renovation), but it was comfortable, convenient, and quiet. It had a great location blocks from the waterfront, within easy walking distance to Salamanca Place, shopping, art galleries, and heaps of cafes and restaurants. Our room had a harbor view.

There are a lot of hotels close to the waterfront around Sullivan's Cove. If you go to Trip Advisor and search for hotels in Hobart, you can find a ranking of hotels based on reviews, and read the reviews. You can also click on a map that will show you locations. Battery Point, which Melnq8 recommends, is adjacent to the central district waterfront area. It is quaint and charming, quieter, has lovely old architecture and a nice neighborhood feel. You could probably find a B&B there.

You should check out the prices on Wotif.com Their site also features photos of the accommodations on offer, so you can get a sense for what the rooms look like, maybe what the exterior or the public spaces of the hotel are like.

My husband and I considered staying one night at Moorilla Estate Winery, which is located in the suburb of Berriedale, 12 km north of Hobart's city center. It's not a convenient location for you but it looked like an interesting place to stay, very modern in a wooded location by the water. There is a museum on site and a winery, of course.
http://mona.net.au/visit/sleep/

I don't remember downtown Hobart as being that frenetic or noisy.

If you end up staying in one of the large hotels near the waterfront, ask for a room on a higher floor, which should be quieter.

I always bring ear plugs, no matter what.

I stayed at Henry Jones Hotel last time I was in Hobart it was very quiet even though it's on the waterfront. I've also stayed at the Casino with a fabulous view up the Derwent River that was also very quiet and very reasonably priced.

There's a free ferry out to MONA now, apparently non-Tasmanians will have to pay an entry fee to the museum from this month. By all accounts it's an amazing private museum, with an art collection assembled by a gambling millionaire, he comes across as a rather studied eccentric.

I'm confused, are Moorilla Estate and Mona related or are they separate places?

They are the same place now. The same guy owns it all, there's a museum and winery and accommodation all in the same place. I haven't been, but most of my friends have. The accommodation is apparently very expensive but the museum is great.

http://mona.net.au/visit/drink/

Thanks for that Susan.

Thanks all! Battery Point gives me another area to look for accommodations. Since we are visitors to the area,it is nice to have help with locations

Hey Other Mel!

I've stayed at Morilla before (loved it) but before MONA - I'll be there in a few weeks and can report back.

Best,

Melodie

That'd be great Melodie.

Great Melodie, we will expect a report soon!

I was at Moorilla before they built the museum, as well. But we loved the winery and the accommodations looked fabulous (if you like modern architecture, which we do). My husband is a big fan of their Gewürztraminer.

Outside of Launceston, there is another winery that offers accommodation, Rosevears Estate. Its setting near the Tamar River is lovely. Unfortunately, it looks like their bistro is now closed, but there is always beautifully situated Strathlynn Restaurant at nearby Ninth Island Winery (which is related to Pipers Brook Winery).
http://www.rosevears.com.au/