What was the primary goal of the battles in Tennessee?

For the Confederacy

To prevent the Union forces from seizing

1 -- Its important railroad net, which was essential for the war purposes. Particularly the railroad line Memphis-Corinth-Decatur- Chattanooga-Knoxville-Lynchburg, most of which running in Tennessee was paramount as communication line for supplies and troops quick displacements between Confederate eastern and western theatre of operation.

2 -- The Appalachian gaps of Chattanooga, Knoxville and Cumberland, whose control would prevent the Eastern Confederate States from being westerly invaded.

For the Union.

The conquest of the line Cumberland -- Mississippi with the seizing of Nashville and Memphis, which would have outflanked and beaten off any Confederate endeavour of invading Kentucky and given the way for further seizing of the aforesaid vital railroad communication line leading eastward to Chattanooga-Knoxville and the subsequent control of the homonymous Appalachian gaps. By so doing the Union would had had the keys to invade Georgia and the eastern Confederate States. Furthermore it would have marked a great progress of "Anaconda Plan" one of whose tasks was that to take control of the whole Mississippi river

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