The Lincoln Museum sits on the town square and tells the Lincoln story through a series of life-size dioramas, period objects, campaign material, and local memorabilia. It is old-fashioned in style, but that suits the town: direct, earnest, and focused on the broad arc from Kentucky cabin to Ford's Theatre.
Visit it after the national park sites if you want the full narrative filled in. The museum is especially useful for families, because the staged scenes are easier to follow than a wall of dates and documents.