Tasmania, Australia transportation and animals

Best to start with official websites like this one but there are many.

https://www.discovertasmania.com.au

It has been many years since I worked there, but I think a car will be the way to go - there are too many interesting side trips to count on public transportation in a time-efficient fashion. For a small place it is quite" large" and things are spread around.

There is a public transport network of buses but renting a car is going to give you far greater options. I will need to look up the bus network to see where it will take you.
The animals.
Bennett's wallabies and pademelons are pretty common in many bushland areas. I have seen plenty of wombats, brush tail possums and spotted quolls while hiking. Also a few echidna.
Fairy penguins can be found in a few places. The Neck at Bruny Island and also in Bicheno.
Devils? You are a million to one chance of spotting them in the wild.
I know of 4 Devil sanctuaries.
Devils @ Cradle near Cradle Mountain.
East Coast Nature world near Bicheno
Tasmania Devil Unzoo near Taranna
Trowunna Wildlife Park near Mole Creek.
There are no wild koalas in Tasmania.
The common Eastern Grey Kangaroos of the mainland (40 of them staring at me right now) are not as common in Tassie only found in a few areas.
On Bruny Island I spotted a white Bennett's Wallaby on 2 occasions. Hiking Cradle Mountain"s Overland Track we often had our path blocked by wombats, wallabies and pademelons. Around campsites the quills would sneak around trying to pinch any food left lying about.

Looks like Tassielink can get you to plenty of good spots.
Hobart to Huon Valley.
Hobart to Bicheno
Hobart to Tasman Peninsula/Port Arthur
Hobart to Campania via Richmond

Redline run between Hobart and Launceston.

Any of the Tasmanian National Parks would be good choices for wildlife spotting, not to mention providing some of the best day hike opportunities in all of Australia.
Some favorites are: Cradle Mountain/Lake St. Clair about 2 hours west of Launceston. One of the Devil sanctuaries that Tintaldra mentions is on the way and is worth a stop. The circuit around Dove Lake is boardwalk all the way - about 1.5 miles as I recall - and you'll usually see a variety of critters along the way, especially in the early morning or late evening. The park itself is worth a few days.
Mount Field NP about an hour and a half from Hobart ... likewise some very pretty hikes thru the Tassie bush with the likelihood of seeing a variety of creatures. It's the first place we saw a platypus on our own travels.
Freycinet NP is about 2.5 hrs. from Hobart along the coast. The hike down to and along Wineglass bay is especially pretty.
I agree that having a car really is the way to go. While not needed in the cities one would maximize your freedom of movement and make the most of your available time once out into the countryside. The driving is not especially daunting - the rural roads are well maintained, two lane affairs, and about the only impediment I can think of is the presence of wildlife at dawn and dusk.

Thank you so much! This is really helpful information. Would you suggest staying in one place for five days? Or two different areas of the island?

I'd suggest flying into Launceston and out from Hobart (or the reverse) in order to keep from backtracking.
Spending a few nights in or near Cradle Mtn. would allow you to get out early to see the animals foraging. They tend to be relatively inactive during the middle of the day - bedded down while it's warm.
The rest of your time could be spent somewhere close to Hobart, making for a very pleasant day trip while crossing the island.

Just a sketch of a plan based on a few of our travels.
Fly into Launceston as Robert suggests. Hire a car, we hired a little Nissan Micra on one trip, did the job.
Drive to Cradle Mountain (you will need to purchase Nat.Park pass) and stay at Waldheim Chalets. This is where we (6 of us) stayed before doing the week-long Overland Track hike.
While staying there I saw wombats, Bennett's wallabies, pademelons, brush tail possums and echidna. As in they were hanging around the chalet.
From there you can do some beautiful day hikes.
2 nights there.
Drive toward Hobart via Derwent Bridge. Spend a night at Derwent Bridge Chalets to break the 5 hr drive to Hobart. You could do a visit to Lake St Clair (the finish of the 7 day Overland hike).
Drive onto Hobart and onto Richmond. Stay in Richmond 2 or 3 nights.
From Richmond day trips to Freycinet N.P,/Wineglass Bay and Bicheno.
Another day trip Port Arthur and Tasman N.P.
It is a short trip from Richmond to the Hobart airport.
Just a sketch but maybe worth some research.
Just remember you do need Nat.Park passes for entry to Tassie's amazing N.P's.

https://www.bonorong.com.au/

This information is super helpful in our planning. Thank you all