If you're going to commit to only one software, I'd personally go with Illustrator, cause I think it's more fun. A lot of the stuff you can do on Photoshop can be somewhat replicated on Illustrator, but Photoshop can't offer the most crucial feature of Illustrator - lossless scaling.

It really depends on whether you ever plan on using scalable vectors. Like if you create some artwork for a coffee mug on Photoshop and then later decide that you'd like to scale it up and also make a poster out of it, you're kinda fucked.

For images like the one on that T-shirt, you could very easily, cleanly and quickly do it on Photoshop - shapes and lines too. As the name suggests, Photoshop is ideal for photos - not graphics. You gotta figure out if you're more keen on photo editing or graphic design.

With artwork like that, as long as you're not planning on scaling up the image, it doesn't need to be a vector. If you decide to go with Photoshop, you could just always start off with making all your artwork massive in size, and then scale it down whenever necessary.