You can absolutely ride the monorail just for fun. If you take a look at the videos found
HERE, and find the one called "Transportation," you'll see me and my daughters having a great time riding the monorail. Speak to the cast member attendant at the monorail station and often you will be able to ride in first car with the monorail driver. You may need to wait for a train or two to cycle through, but in my opinion, it's well worth the wait. The view from the first car is incredible. Plus you get that cute monorail driver's license as a souvenir.
My family often skips the parks on our first day at Walt Disney World. Having a character dinner at a resort, like Chef Mickey's at the Contemporary, is a great way to get in the Disney spirit. Another character dinner choice would be 1900 Park Fare in the Grand Floridian (here you'll see Cinderella, Prince Charming, and Cindy's stepmother and stepsisters). Having a character meal really makes you feel like you're on vacation, but you don't have to expend lots of energy like you would if you were running around the parks -- perfect for a day when you may be tired from traveling. You are also welcome to tour the resorts, many of them have incredible architecture or amazing themeing that are well worth the trip.