For a very pleasant way to spend your day take a walk through Showa Kinen Park (Showa Memorial Park), which was established to commemorate the 50th anniversary of Emperor Showa's reign. The park is a pleasant place to stroll and if you go be sure to visit the Japanese Garden and see the bonsai. If you're lucky, the gentlemen who take care of the bonsai will be at work and they are friendly fellows, willing to answer questions about the bonsai and the care of bonsai (in Japanese only).
If walking is not your thing you can also bicycle through the park. You can either take your own bike or rent one there. The fee for renting a bike for three hours is 400 yen for adults and 250 yen for children.
There are plenty of activities for the whole family so you can plan just a few hours there or spend the entire day.
In addition to bicycling you can enjoy barbecues (take your own supplies or buy there - but take note, you must reserve in advance), pedal-boats on the small lake (pond), putt-putt golf, disk golf, and various other lawn games. A full list of activities and prices is on the website.
If you're not inclined to avail yourself of the various games you can just enjoy the greenery, relax in the shade with a book or your music, do a bit of bird watching, or have a bite to eat at one of the food areas.
Children will enjoy the Children's Forest. Big kids (the adults) like it too.
If you're going on the weekend expect the crowds. Like most popular family friendly places in the Tokyo area, Showa Memorial Park draws a very large number of people on weekends and holidays when the weather is nice.
Getting to the park is relatively easy. You can take the JR Chuo line train from Shinjuku to Tachikawa Station. The Akebono Gate of the park is about a 10 minute walk from here. If you ride the JR Ohme line get off at Nishi Tachikawa Station and go to the Nishitachikawa Gate, about two minutes from the station. Other train directions are shown on the park's website.
If you go by car parking is available for 800 yen and the three parking lots can hold about 2,500 cars. These lots fill up pretty early on the weekends, so if you're driving, leave early.