A whale or nature cruise. Bar Harbor may not be the trendiest place to see whales, and it sure isn't the warmest, but the companies that run their are pretty good and work hard to make trips memorable. I would recommend any that advertise a College of the Atlantic or Allied Whale connection, this is a local research base that has up to date knowledge of the whale populations in the Gulf of Maine.
Acadia National Park is a must if only driving around the park loop road. It was set up by the Rockerfeller family who had the foresight to ban cars (other than the loop road). However there is an excellent network of carriageways for walking and cycling. Sand Beach is a great destination for a sunny day - take your own picnic and watch the waves, Jordan Pond is a good stop for refreshments (moderately priced), gifts and a walk around the lake. The top of Cadillac Mountain at sunset/sunrise is magical. I was truly in awe of the views over Frenchman's Bay and further out to sea, and worth persevering even if foggy as it may be clear at the top.
Quick Tips:
Eat at Miguels Mexican Restaurant, but make sure you are hungry. The margaritas are the best.
You have to go to Ben and Bills Chocolate Emporium -- ice cream and chocolate heaven.
Have a lobster meal -- try it if you haven't had it before. Some of the smaller villages by the harbours have a less commercial feel.
Go sea fishing or on a lobster trip -- you might get your own fresh lunch.
Visit the Visitors Centre before you get into Bar Harbor -- they have lots of information and leaflets -- a good orientation before your visit.
On a rainy day the College of the Atlantic, Eden Street, has a small but interesting natural history museum.
Acadia is a small but wonderful park as is the whole of Mount Desert Island so taking time to discover some of the more out of the way places will no doubt reward you.
Best Way To Get Around:
A car is great as you can explore at your own leisure. Walk around town - you don't need a car, in fact it is probably a hinderance. Cycling in the park is a great way to get closer to nature.