The first thing to do is find out about the walk you are going to undertake. This website can help with some, and there are many guidebooks. Other walkers are a good source of information, as are maps of the area. You can use other websites, or contact the Parks and Wildlife Service at Huonville or Geeveston.
You need to know about your main walk, and any sidetrips or options. Write down your walk plans, details of the vehicle you are in, and include the time you expect to return. Give this to someone reliable. If you are touring, accommodation operators can help with this, as can local police. You must remember to confirm your return.
Note that mobile phones work from many mountains which are regular bushwalking targets. Depending on your philosophy, they can be a useful safety tool. CDMA phones may be more useful on some of the more remote peaks, but GSM phones work from the closer ones. However, you shouldn't rely on them - what happens if you get injured somewhere the phone has no reception? Remember that sometimes phones will send SMS messages when voice calls are not possible.
Check the weather. Poor weather in Hobart will generally mean poor weather on the mountains. Could be a good day for a walk in the forest, or to a beach. Good weather in Hobart will not necessarily mean good weather on the mountains. Use your own judgement, but remember the weather can change very quickly. Read about Tasmanian weather here.