My boys love to visit the North Carolina Museum of Natural Sciences. Their favorite subject is science. We have visited this museum a few times in the past year. There is evolution posted on many of the exhibits, but since we are a Christian family, my boys have learned that evolution is not true and we are able to discuss the different view points in each exhibit.
Another free museum that is right next to the science museum is the North Carolina Museum of History. This is another place that one could spend hours visiting. We usually visit one museum in the morning, have lunch, and visit the other museum in the afternoon. There are only two stories that are accessible to visitors, but there is so much information on the history of North Carolina. This is a great place to visit when your child is learning about North Carolina history. The main exhibit on the first floor, tells about the history from the beginning (Native Americans) through the 20th century. There are a lot of hands on activities throughout the exhibit. One of the other exhibits that my kids like to visit is the sports hall of fame. Here is a map of the exhibits.
Both of these museums are open from 9-5 Monday through Saturday and noon-5 on Sundays. Parking is only free on the weekends.
A couple more places to visit in the Raleigh area would be the State Capitol and also Marbles Museum which offers a discount to homeschool families.