We will first arrive at the airport (HBA) in Tasmania, planning to go to the Sorell Fruit Farm, Marakoopa Cave and the Tasmanian Chairlifts.
We can't rent a car because we are still under 25, so how can we get there? What kind of public transportation is available? Is it convenient?
It's hard to travel in Tasmania without a car -- but you can most definitely rent a car if you're under 25, although you'll need to pay a premium. Check out Redspot, which serves both Hobart and Launceston airports and happily rents to drivers 21 and over, plus random small local hire companies (Google or the first hostel you stay in will have listings).
If you're not uncomfortable with a little uncertainty, hitch-hiking is easy, and reliable almost anywhere in Tasmania. Folk are friendly and as a wilderness state it's been the custom to hitch into and out of bushwalking tracks for generations.
A modern fear and paranoia exists which will decry the suggestion, recommend against it, and possibly denounce me for suggesting it. But as one who has practiced it regularly and offers rides regularly, I can caution against lending these fear mongering conservative elements too much of your ear.
The real question is whether you are comfortable with uncertainty as you can't be sure of when you'll get places or if and how. Mostly hitch-hiking is very reliable and can get you from A to B faster than public transport, but if you're free and easy and open ended, sometimes you might end up changing your plans a little along the way, and accepting an invite to tag along with someone, or worse get stuck without a ride for longer than you like.
In Tasmania it's fair to say my average wait would be 15mins and the longest in a decade has been perhaps an hour or two.
Failing that try: https://www.shareyourride.net/carpool/Australia
Tasmania does have a city bus network to take you places. There are no trains. Taxis are also only available in the towns like Hobart. To travel cross-country, you can pick from a multitude of bus lines.
