I will give my top ten, but remember I like uniforms and such so my list might be bias.
1. Gettysburg National Military Park
2. Battle of Chickamauga, the muskets and rifles collection is impressive, the battlefield is interesting
3. Antietam National Battlefield, for some reason this is one battlefield I really enjoyed, perhaps there was a certain sad feeling there.
4. Bull Run/ Manassas, two battle for the price of one.
5. Shiloh National Military Park
6. Vicksburg National Military Park
7. Museum of the Confederacy
8. Tennessee State Museum, great uniforms , I was allowed to tour the storage area and view the Confederate battle flags.
9. National Civil War Naval Museum, but only if you also spend a day at the nearby Infantry Museum
10. Civil War Museum Kenosha Wisconsin, You walk through the War and get to "interact" with motion activated Civil War manikins. Also you get to stand in the center of a battle and have a 360 degree view.
Some others can make the list: Appomattox Courthouse National Historic Site, this probably need to find a spot on the top ten. Battle of Kennesaw Mountain, you get to see the mountain and the view from the mountain is great. Pea Ridge National Military Park an ancestor with my last name won a Medal of Honor and promoted to general, so I had a personal connection. Harpers Ferry, important site, a lot to see and beautiful place.