With a small boat like yours you can anchor in pretty much every harbor as Tom Young says. If it doesn't look like you can anchor, try hailing the harbor master. If you don't get any response on the VHF just take an empty mooring and go ashore and ask around. The only time this is a problem is when you are in some of the small harbors with predominantly lobster boats.
Here are a few pointers for harbors around Mount Desert Island.
Northeast Harbor - Hail the mooring agent on 9. They will tell you which mooring to take and will generally come out in a skiff and guide you to it. Northeast is very crowded and pretty yachty, but is a good harbor for doing Acadia National Park.
Southwest Harbor - Officially you aren't supposed to anchor in Southwest Harbor, but there is room in a few spots on the south side of the harbor. The town has 5-6 rental moorings clearly marked "Town Rental" in the middle of the harbor. If one is empty, pich it up. The harbormaster Adam Thurston will be by to get the fee (around $20-$25) or you can go to the town dock on the south side of the harbor by dinghy and pay him there. Hinckley also rents mooring in the outer harbor ($35 or so). Hinckley moorings include use of their showers and laundry. If you take a town mooring you can go ashore at any of the three town docks. If you are feeling flush, you can go to the Great Harbor Marina - hail them on 9.
Islesford (Little Cranberry Island) - Thre are several free moorings labelled "Town". They are all at the outside of the mooring field. Just pick one up if it is empty. There is also another free mooring further in labelled "Kitsy Memorial - red lettering on white ball). Just pick it up if it is empty. Go ashore at either of the two docks. You can also anchor outside the moorings but the bottom has a lot of kelp so make sure you are well set before you leave the boat.
Great Cranberry Island - Pick up one of the free town moorings (labelled "Town") at the outer edge of the mooring field. Go ashore at the town dock. If you walk up the main road about 1/2 a mile there is a nice little cafe. There is also a nice trail to a pleasant beach that starts at the cafe.
Somes Harbor - Anchor anywhere there is room. You can go ashore at the prominent dock on the left side of the mooring field.
Valley Cove - You can anchor in rocky bottom around the edges. There is lots of current here, so make sure your anchor is well set before you dinghy ashore to climb flying mountain.
Bass Harbor - You can anchor in the outer harbor on the west side, but you will have a LONG dinghy ride to get ashore at either the town dock or the ferry dock. You can also rent a mooring from Morris Yachts on the east side of the outer harbor. I would only stay there if it is calm or blowing from the north or east since it gets lumpy when the wind is from the southwest.
Frenchboro - Pick up an empty mooring in the outer harbor. If they are charging again, pay at Lunts dock on the right side of the harbor when you row in. You can get lobster and other food at Lunts and a couple of hundred yards further along the road at a second cafe. Frenchboro has LOTS of trails all over the island.
Burnt Coat Harbor (Swans Island) - Either anchor where there is space or take a mooring with a green pickup buoy. You can pay at the lobster coop if someone doesn't come around to collect. To go ashore tie the dinghy up at the lobster coop - ask where to put the dinghy.
Mackeral Cive (Swans Island) - Anchor anywhere you want in 7-12 feet at low. My favorite spot is on the east side of Roderic head. You can also anchor by the ferry dock where you can go ashore. There is a restaurant up the road a bit from the ferry dock.
Buckle Harbor - Anchor anywhere. Go ashore at the NE end of Buckle. It is best to go ashore at half tide or more. Thre is a nice trail around the island.
Blue Hill Harbor - Pick up a mooring at the Kollegewidgewok (sp) yacht club or anchor in the inner harbor. You can go ashore at the yacht club or at the town dock in the iner harbor (high tide only unless you like dragging the dinghy through knee deep mud).
Bar Harbor - Pick up a town rental mooring and pay at the harbor masters office at the town dock. Enjoy the ambiance of thousands of tourists and be lulled to sleep by the gentle roar of cruise ship generators. You may have guessed that I don't care for Bar Harbor. On the plus side there are LOTS of good restaurants there and a Hanifords grocery store within walking distance of the town dock.
Sorento - Thre are a couple of free guest moorings at the edge of the mooring field.
Hope this helps