It would help to know more specifically where you are in the east bay. There are three primary riding locations in the east and south bay. All have designated areas for kids and beginning riders. But each also have their pros and cons...
Carnegie OHV park is the farthest east and is located between Livermore and Tracy. Carnegie is known for its steep and very technical trail riding. People either love or hate Carnegie for this reason. If you stay down in the flat areas there are designated tracks for kids (engine size restricted) and intermediate riders. Carnegie was once the site of an old brick factory due to the high clay content in the soil. During wet months the clay can turn to a very thick mud and is not be desirable for riding, although some will try. During the summer Carnigie can get hot and very dusty.
Holister Hills OHV park is the farthest south and is located in the hills above Holister (I guess the name is fitting). Holister also has designated kids and intermediate riding areas. The majority of trails at Holister are well groomed, designated one-way, and in general are less technical than Carnegie. Holister has a high sand content in the soil so during wet weather the mud is much less sticky than the mud at Carnegie. Depending on your location it can be a long drive, but the town of Holister has some nice places to eat if you aren't in a rush to return home.
Metcalf Park is just south of San Jose, so it is fairly close. It is however much smaller than Carnegie and Holister Hills. It is run by the county, so California OHV season passes are not accepted. It also has designated areas for kids and beginner riders. The trail system is small, but does offer a variety of terrain for more experienced riders.
All three parks charge a daily fee of around $5, and you can purchase a California OHV season pass for $50 that is good at Holister and Carnegie.
Most important is to be safe and have fun
