TL;DR Yes you can see most iconic Aussie animals in Tas (except koala, emu or crocs) and I would argue you can see those wildlife much more easily in Tas than other parts of Aus.

There aren't really zoos in Tas but there are lots of wildlife sanctuaries, rescue and rehab centres, and breeding centres for the Devils who also house habituated Aussie and specifically native Tassie fauna. There is a place on the north coast that breeds echidna and platypus. I highly recommend trying to see the native wildlife in the wild and then visit Bonorong near Hobart if you haven't ticked off your list by the end of your hols, both for their up close experience with roos and devils and because they have an excellent ethos and wildlife rescue program.

You will see Bennett's wallabies and pademelon throughout Tas if you stay anywhere in a bush setting or small town, or if you drive at night (so please drive slowly). You can see wallabies and pademelons in the day in the bush, for example around kunanyi-Mt Wellington lower slopes by Cascade or Fern Tree or walking aroung Freycinet. There are many smaller bouncy marsupials that I have only seen at night, like potaroos, bettongs and bandicoots. You should see brush tailed possums if outside at night too, especially in camp sites.

Eastern grey kangaroos tend towards the north, east coast or central grasslands. Mobs can be seen (and got close to) at narawntapu or wukalina-Mt William. They also occur in smaller numbers all along the east coast highway and Maria Island.

Platypus are surprisingly common in Tas but shy. They have been recorded in Hobart Rivulet within 1 km of the city centre! But your best chances are Mersey River at Latrobe (warrawe forest), Tyenna River at Mt Field to Westerway, or an accommodation with a dam or river with residents (they are territorial). Platypus females are easier to see in summer when they will feed through the day to ensure they can feed their young. Be very quiet and still.

There are particular accommodations that feed wild devils and quolls. Devils may also be seen throughout Tas on rural roads at night but they don't appear close to roads that often. The best bet for quolls seems to be Mt Field or the north-west forests, but I've not seen a wild one in 2 years of living here.

I've seen most echidna on the East Coast (keep an eye out for them on the highways and slow down!) and on Maria Island.

Wombats are common on Maria Island and in many places with grasslands, even on the top of mointains like Mt Field and around Cradle Mountain.

There are also plenty of cockatoos and parrots, eagles, whales (in the right season) and penguins (Bicheno, Low Head or Bruny Neck are best) as well as snakes! With a bit of bush walking (of which there is plenty) and a few overnights in small town or bush settings, you should see a lot.

We just don't have koala or emu (there used to be a Tas emu but alas that was killed off during european colonisation, like the thylacine).