During the Devonian period (410-360 million years ago) warm, shallow seas covered parts of Ohio, especially eastern Ohio. The area had shifted north and then existed around the Equator during this period. The picture included below shows the topography of Ohio and location of the shallow seas during the Devonian period (image source). The shales that cover eastern Ohio were created from the sediments that deposited in this period.
In the late Mississippian period much of Ohio was once again covered by shallow seas. These warm, shallow seas deposited lime sediments that are seen at the surface of northwestern and east-central Ohio. As for the surface rock in Springfield (in central Ohio), much of it was created during the Devonian period. The rocks at the surface are composed of the sedimentary rocks limestone, dolomite, shale, and sandstone, which deposited from erosion.
In summary, for much of the geological record Ohio has remained in tropical climates. This tropical climate has impacted the conditions under which the sedimentary rocks in Ohio have formed. Plate tectonics have shifted Ohio northward only in relatively recent geologic history– by the Quaternary period most of Ohio was covered by glaciers.