There's three main routes to get to the East Coast. A4 from Campbell Town in the west, A3 from Hobart in the south, and A3 from Launceston in the north-west. All take about an hour and a half.
TassieLink provides bus services to some towns, but not all.
Like most of Tasmania, there isn't much public transport in this area, and your options are limited to by car. However, on the contrary, East Coast does have a good network of roads and highways albeit those roads being winding and slow, which is comparable to those on the New Caledonian mountains. However, the primary routetakes you to most of the towns around here.
If there's one thing that is not to be missed on the East Coast, that is wildlife, in particular note, penguins. Yep, while most people associate Tasmania with the Tasmanian Devil, the East Coast is more popular for its penguins. Bicheno has penguin tours that run nightly. Some notes with penguins; Stay still, don't shine a torch on them or use flash photography. They have their route from the beach, and will stop or retreat if you are in their way. Back off to the side slowly. Control children.
The beaches in the East Coast are also nice too, and a bit different from the mainland. Be aware though, that it can get cold, but a jacket will largely do the job.
The two most done recreational activities done here are hiking and fishing, which the East Coast has lots of opportunities to offer for both.
Seafood on the East Coast is quite popular as one of the local industries here is fishing.
Most of the bars and pubs here are just like an ordinary Australian, except that. they never get violent or anything like that here, but rather the opposite.
The East Coast offers ample opportunities for both camping and staying at a motel, so it's your choice.