I agree with most everything bluechip508 has said except I disagree about the Lobster Pot. Having gone to P-town to stay for a week is something I've done as an off season activity for over 2 decades.
When I've met friends there, they always want to go to "The Lobster Pot". I have finally refused to take people anymore. I've never had a good meal there. It is, to me, the quintessential tourist trap. There are so many other places to go. If you want regular food that's good, ask the people who live there, they all work behind the counters of one of the many stores. I'll take the fried clams (whole, the way God meant them to be eaten) at the Mayflower cafe (an old style 'diner') in a heartbeat over the Lobster Pot.
There's a lot open and a lot closed off season but if your looking for activities, with moments of quiet, stay on the west end of town. It's quieter and nicer to walk around.
The P-town gym is a little better to work out in if your more focused on that (actually working out) vs. Mussel Beach which is smaller, more expensive and 'cruisier'.
Hiking the fire trails on the Provincelands (other side of route 6 right near the end of Shankpainter Rd) is great and you can get lost in there so make sure the GPS on your phone works or get a map. It's a great mix of woods, ponds, and dunes.
Most galleries on the east end tend to be closed during the week on off season, so if you want to check them out, find out when they're open as soon as you get there. As bluechip508 said, there's some great art to be seen and if you've got the money, bought. Keep in mind though with the art there, you get what you pay for. A 'painting' for $150 is probably worth about $20, but the more pricey works tend to be actually better art (not just because they are more expensive, but because the tourists who are looking for 'deals' in the galleries don't know anything about art. But there are quite number of real collectors and connoisseurs who know the difference and they shop there too, but know that the cheap stuff is just that, cheap and bad.
Restaurants open and close all the time there. There's no shortage of them even in the off season. I too (like bluechip508 have eaten well at Fanizzi's (many times) and Ciro & Sal's (but I'd choose Fanizzi first) as well as the Mews (more expensive but well prepared and served). There are little places for regular inexpensive food like the Post Office, or the Portuguese Bakery (open all year but not a 'sit down' place), Haven has good breakfast and lunch. Also Rossi's Grill (second floor of Whalers Wharf all the way in the back and overlooking the bay) is good for lunch and dinner. Avoid the restaurants with 'hawkers' in front, they're always bad if there trying to pull people in off the streets. There's Napi's near the center of town (Standish and Bradford St.) for lunch or dinner (I don't know about breakfast since I've never had it there.)
I've several friends from the Boston and CT areas that have moved to P-town, and even with a whole week in the off season, I've actually been too busy to spend time with them (insensitive of me, yes, but I get caught up in all the things to do.)