Indian Riverside Park is a beautiful, large park on the intracoastal waterway with the wonderful Children's Museum of the Treasure Coast, a splash pad, river boardwalk/loop trail, pier, white sand beach, mansion, captain's house, mangrove boardwalk, and path around a pond with ducks. The park has lots of shade, and the beach on the intracoastal waterway has sugar white sand and mangrove trees- the water is brackish. A large tree shades the white sand beach and families picnic here while the kids splash in the river (I'd beware the water quality though). The splash pad doesn't have any shade but is colorful and looks like it would be fun on a summer morning. A pier reaches out into the blue waterway and there are picnic tables with pretty views.

There is a cute dolphin fountain at the entrance to the park. Nearby is a pond with a path around it. The pond offers lovely views of the cypress trees, blue sky reflecting in the water, ducks, ibis birds, and cypress knees along the pond's banks. 
The Mansion at Tuckahoe sits on a tall hill, Mt. Elizabeth, in the middle of the park. It is white, in Mediterranean style, and quite impressive. An amphitheater on the river sits below it, and is a spot for winter concerts for the public, Music at the Mansion, and Winterfest, a family-friendly event. The mansion was built in 1938 by Willaford and Anne Leach- she was a Coca-Cola heiress. It was the setting for many parties and quite the local social scene. You can have your wedding here and enjoy the huge terrace! The public can tour the mansion, on the first and third Wednesdays of the month at 10am or 11am from Oct to May, but you must email first to reserve a spot.
Past the mansion is another house, Captain Henry Sewall's house, by a mangrove swamp (it was rescued from demolition in 2006 and relocated here). It is a sweet house with a porch, built in 1889, where he lived with his wife Abbie for 21 years. Captain Sewall, from a prominent shipbuilding family in Maine, came to Florida after traveling the world in his family's sailing ships (he was one of only four survivors of a shipwreck off North Carolina in 1868). He built his home on the tip of the Sewall peninsula and in 1891, constructed a long freight dock and post office, which was also his house. Mail boats and everyone traveling down the Indian River stopped there. He led the movement to dig the St. Lucie inlet in 1892, which allowed boats to access the Atlantic Ocean, and salt water fish to enter the estuary. He spent his life in politics: as post master, commissioner, and in the legislature, representing this newly-settled part of Florida.
You can walk in the mangrove, behind Sewall's house, on a wooden boardwalk and look at the fascinating mangrove trees. The sunlight comes through the trees, and hits the green-brown water. The boardwalk ends at a picnic pavilion with a thatched roof. 
Afterwards, have some lunch at Jensen Beach Boulevard, enjoying the breezes off the water. Another great spot for lunch is the Drift Kitchen at Hutchinson Shores Resort, with wonderful ocean views. If you don't have time for those, you can stop at Jennifer's Bakery on Ocean Blvd for a cup of coffee. 
For another lovely park with a playground with tall, tall slides, check out Sewall's Point Park, a five minute drive south on Sewall's Point Rd.

Directions

Indian Riverside Park is located at 1707 NE Indian River Dr, Jensen Beach, Florida 34957. There is plenty of free parking.

Click on map for interactive view