If you’re printing a picture, do not print them with a watermark on them. If you’re delivering a high res image to someone who has purchased the digital file, do not give them a watermarked image either. Watermarks should only be used when your photo is displayed online to the general public.
Adobe Illustrator is an extremely advanced program that is used to create vector illustrations. Some people use it to make logos. It will not help you add a logo to an image though.
The best way to quickly add a watermark to an image is to use Lightroom. Here’s an article that explains how to do it well (along with a lot of other good watermarking tips): https://photographylife.com/how-to-watermark-images-in-lightroom