Cherie Down park is about 1 and a half hours away ( past PC) and offers free parking. You walk across a footbridge and are right on the sand, this is our favorite beach in Cape Canaveral and it is very beautiful. Pack a picnic or plan on getting some to go at the area McDonalds which is right on the way to the beach. There are a lot of other places to eat once you are in PC but if you want to sit on the beach and enjoy the fresh air, Mickey D's has a drive thru. lol
Leu Gardens is very nice with many pathways that take you around the gardens. It is about 30 minutes on I4 to Princeton to get there. More of a side trip kind of place if you like looking at planned gardens (they have an amazing rose garden in season) There are many paths to take though and is a very beautiful slice of tranquility in the middle of Orlando.
Moss Park is off of the 417 in Orlando and offers wonderful trails and some educational programs that you might be interested in. It is about 20 to 30 minutes away from Disney. Take 417 to the Moss Park Rd exit and head due east the park entrance has a guard that can give you maps and information.
Blue Springs State Park is incredible and about an hour drive up I4 to Deltona to get there, you would get off I 4 and then travel a bit longer to get back to the Springs. The water is always 73 degrees and this time of year, the manatee activity is incredible.

Another of our favorite places to go, although we prefer to go in summer when the cool water is refreshing in the heat of the day.
If you want to venture further out of the Disney zone, you could go west to the Gulf of Mexico beaches. About 2 hours to get to those but there is one I absolutely love. It is called Caladesi Island in Dunedin, FL . You park at Honeymoon Island and take the ferry boat over to Caladesi. You land at a hospitality house and then walk down the boardwalk to the beach. You can rent umbrellas and chairs if you want, but this time of year the sun is not nearly as scorching as it is in March to September. The water on the Gulf side is clear and a bit warmer than that of the Atlantic Ocean. We love the natural beauty of land and walking along that beach is so wonderful.