Not all Tennessee counties hold primary elections, and some only have a primary for a single party. I have checked a few election commission websites, and it looks like tomorrow (Wednesday, April 16) marks the start of early voting in most places that are holding primaries. Figuring out which locations are open on which days during early voting can be a little trickier.
For example, in Maury County, the election commission office in Columbia is open every day during the period except for Good Friday. The Spring Hill site is only open on the last three days of early voting. And in Shelby County, the downtown location is available throughout the early voting period, while satellite locations open on the 25th. Some counties’ Saturday hours are shorter than others. And so on.
If you are not sure where to vote, or whether your county/party is holding a primary, contact your county election commission. The Tennessee Secretary of State’s office publishes a directory online.
If you don’t really belong to a party, you might consider voting in the primary election anyway; else you might miss a random general election, for example the Chattanooga City Court Judge positions here locally. Both judges are running unopposed, but still. Election Day for the primaries is Tuesday, May 6; on election days you vote at your precinct’s assigned polling place, and NOT at an early voting location.
The general election for every county will be held on the same day as the state primary elections: Tuesday, August 7. You definitely won’t want to miss that one, as it offers you more levers on your government than any ballot in the past eight years. And (fingers crossed) you’ll have a brand-new tool to help you prepare for it.