Using Adobe Illustrator is the Best way to change the color. Because then the image is still vector and can be enlarged to your needs. In Photoshop, if you raster the image then you can not enlarge. But you can place the image as a vector in Photoshop, still you will need illustrator to change the color.
For t-shirt or any apparel, you give vector art in black and tell printer what color you want to print. They need only black artwork to make screen, they choose the ink color when the print or use Pantone solid colors.
For digital print you will need color image, because here it’s like printing a photograph.