How can I get multiple text boxes aligned and spaced out evenly like my picture below in Photoshop?
Photoshop wouldn't be my first choice to create something like this.
It would be much easier using vector software such as Adobe Illustrator or Inkscape (which is free).
Enable the grid and snap to grid
Create a rectangle, type text in it, centre align the text horizontally and vertically within the box
Group the box and text. Duplicate the group, and use snapping to position the copy.
Repeat the duplications until you have enough boxes
Finally type new text inside
Quick example made using Inkscape
You can export images as PNG from Inkscape, and open it in Photoshop, or import SVGs directly into Photoshop. If you use Illustrator, you can import AI files directly into Photoshop.