The national and state park systems are excellent places to spend a day with a date. They usually have a variety of options that you can choose from to spend your time.
Many have displays of significant nature to the area. In addition, many offer guided tours. Or you can wander around the area on your own and explore to your hearts content.
Fees for national and state parks are usually low and the money goes to the upkeep, preservation and improvement of the area. If you enjoy the park system a lot, many times you can buy yearly passes that will let you into the parks as many times as you want for one set price.