[ghaff]

Backpacking is very different from town to town (or even hut to hut). There are a few places in the US you can go hut to hut (normally). Your options to go town to town are very limited (as in, I don't actually know of any).

In the UK (other places too but the UK is really known for it), there are a ton of town to town walks and you can have your luggage transported.

As for backpacking, as others have said, you really probably want to get started with an out-and-back backpack, possibly with day hike(s) from a base camp rather than setting yourself on a probably overly ambitious point to point week-long backpack. ADDED: People do just strike out to do the AT or the PCT (see Wild) but it's not really recommended.