Tasmania's weather is notoriously fickle, and can be miserable at any time through the year. Tasmania lies in the path of the 'roaring forties' and it is usually pretty windy during spring and autumn. Climbing pretty well shuts down over winter, although occasional fine and still days allow climbing at areas such as Coles Bay and the Gorge in Launceston. Although not indespensible, a car is very useful in Tasmania. Either hire one in any of the cities or bring one over on the ferry.
Buses run between the cities a couple of times a day.
Hitch hiking is a viable option and lifts are usually easy to come by on the main roads, but not so easy on smaller roads. That of course depends on what sort of climbing you want to do. The best sport climbing in Tasmania is at Adamsfiled, Mt Brown and Hillwood. For trad cragging with easy access the Organ Pipes, the Cataract Gorge and Ben Lomond are good. If you want to mount a major expedition to climb big alpine walls then Frenchmans Cap, Mt Geryon and Federation Peak are all options.