The Tasman peninsula has an absolute abundance of dolerite. Although a lot of it is choss, there is plenty that is good quality. From the amazing sea stacks of the Totem Pole and Candle Stick to the modern sport climbing of Mt Brown, the area has cliffs everywhere. There is huge potential for as much climbing as anyone could want, the only drawback are the long approach walks to the undeveloped areas. Despite this, the development of many new sport climbs at Mt Brown and the climbing of the Free Route on the Totem Pole has seen a resurgence in climbing on the Peninsula. New development has also been going on at Cape Raoul. Cape Pillar and Tasman Island have had quite a few routes put up over the years, but details are sketchy.
Port Arthur is a bit over an hours drive from Hobart. The Candle Stick and the Totem Pole can be reached by an hours walk to Cape Hauy from Fortescue Bay, which you turn off to a copule of kms before Port Arthur. Mt Brown (actually Dauntless Point) is half an hours walk south from the Remarkable Cave car park, just south of Port Arthur. After walking along the track for 25 minutes you come to a sandy creek. From here follow the coast around, first passing Parrot Shelf Cliffs, and then coming to an amazingly steep and juggy (for dolerite) cliff covered with eye bolts - this is the Paradiso.
Long and short dolerite sea cliffs. The rock is quite different from alpine dolerite because it has been weathered by the waves, not snow and ice.
There is good but expensive camping at Fortescue Bay and you could also possibly camp on the flat areas on the way to Mt. Brown (but there is no water). The area has recently been declared a national park so entry fees now apply.
There are some great coastal walks on the Peninsula that take in some spectacular scenery. Cape Raoul and Cape Pillar are great, and the walk out to Cape Hauy to contemplate the singularity of the Totem Pole is a must even if you don't plan on attempting the thing. There are also some funky caves and other natural features that don't require walking, including Tasman Arch, Remarkable Cave, and the Tessalated Pavement at Eaglehawk Neck. The Port Arthur historic site is a grim reminder of Tasmania's history as the penal colony of Van Dieman's Land and is worth a look.
Totem Pole (M5 or 25 free), The Candlestick (16), Aquaphobia (22), Super Charger (23), Thunderbirds Are Go (23), Offender of the Faith (24)