I'm pretty new to Arkham Horror and had a quick rule question.

If at the start of my turn I'm engaged with an enemy and as my first action I successfully evade it and move it from my threat area and place it at my location exhausted.

If my second action is to move to another area, am I correct that the evaded enemy stays exhausted at the location it was placed and does not follow me?

Welcome to Arkham Horror!

You are correct, when you successfully evade you exhaust the enemy and disengage from it, so it is no longer in your threat area. Then, when you move, it does not move with you but remains at your previous location.