The Tallahassee Museum of History and Natural Science (formerly the Jr. Museum) is a great place to get to know Tallahassee better. They have three main sections: the historical area, the wildlife trails, and the educational buildings. The wildlife trails are probably the most interesting to both adults and kids, as you get to come in close proximity of indegenous animals such as ocelot, the grey fox, the florida panther, american aligators, golden hawk, white-tailed deer, and black bears. They also have a feature exhibit that usually houses animals that are on loan from another zoo or museum (i.e. penguins, baby tigers, monkeys, etc). The historical exhibit is a functional farm that displays how people in the area lived hundreds of years ago. There is also a much smaller exhibit about the indegenous Native American tribes that lived and migrated through the Tallahassee area. The educational buildings are geared towards younger children - giving them a chance to interact with the nature and history at a more physical level (touch, roleplay, etc) I know my favorite room when I was growing up was the bird watching room/aquarium where you could touch living sea creatures that you might find off of the nearby Gulf Coast (starfish, sanddollars, etc). These educational facilities are also the core of the museum's summer day camps - where children get a deeper understanding of the Tallahassee area through history and natural science. They also have a snack bar and a gift shop.
The admission fees are listed on their website (http://www.tallahasseemuseum.org/) and do not exceed $8. If you have a family with children, I would recommend becoming a member for $50/.year - which gets you unlimited visits, a discount, and some guest passes for your family. They host a variety of community and fundraising events throughout the year (i.e. Jazz & Blues festival, Market Days, Folklife Festivals, and Native American Festivals).