Elk Neck State Park is located eight miles south of the town of North East, Maryland. It's situated at the end of Route 272 on a peninsula formed by the Elk and North East Rivers where they flow into the Chesapeake Bay.
Elk Neck is about 2,200 acres and is made up of woods, cliffs, sandy beaches, and marshes. There's twelve miles of trails in the park, the most popular being the 1.5 mile, easy loop trail to Turkey Point Lighthouse. Another nice hike is the three mile trail out to White Clay Cliffs overlooking the bay. During your hike you'll probably see deer and if you're lucky, maybe a fox or a Bald Eagle.
There are several boat launch ramps in Elk Neck State Park, located at Rogues Harbor. If you don't have a boat to launch, this is still a good place to watch huge barges, canoes, motorboats, and sailboats go by. There's a small store with marine gas, snacks, and supplies, open in the summer. When the store is closed you can fish or crab from the docks here.
If you're looking for a place to swim, Elk Neck State Park has a place for that too. There's a sandy beach with a picnic area where you'll find lots of picnic tables and grills. This is usually a really nice place to swim and cool off, but last summer the area was covered with a mass of green seaweed and slime. There was a sign explaining why, due to weather, currents, etc.. I don't really remember to be honest.
There's 250 campsites at Elk Neck State Park, 15 small cabins, and lots of tent sites. Some of the sites have electric, water, and sewer. There are bathrooms and hot showers available for campers, something I didn't care about when I was younger, but always look for now.
MD Dept. of Natural Resources site for more about Elk Neck State Park.