First take a trip to the local library or bookstore and get a book on fossils for your local area so that you will know what they look like and where to look. Fossils do not occur everywhere on the surface of the Earth but they are common enough that you can probably travel somewhere not too far from home in search of them. Fossils are easier to find in rocks like clay, limestone, sandstone, and shale. Here are some typical fossils, these are from the Missouri Plate.
You are more likely to find invertebrate fossils like these clams corrals, starfish, or fossil plants, rather than fossils of animals with bones--vertebrates.
Once you find a few fossils you can begin to gather more information about what they were like during life, and how they are related to living plants and animals today. Try finding pictures of your fossils in library books or by searching the web.