How do I turn my logo into an EPS file in Photoshop?

For the best quality and to get a file that can be scaled up without loss of quality, recreate the text in Illustrator, not Photoshop. Once you have done this you can save the logo in eps format from Illustrator. If matching the sizing/spacing is critical, place the bitmap image in Illustrator and recreate the text over the top so you can get it as close to identical as possible. Delete the placed image before the final save.