Degrees aren't necessary, but they certainly speed things up. I remember having my mind blown when I found out about what vector files really are several years ago, and that's something you learn pretty much in your first semester in school.
Photoshop isn't necessary but you can accomplish much more with it than without it. Depending on the route you go, you can use crazy stuff to make your cards happen. It's the end product that matters.
For example, you can choose the route I took, which involves Photoshop and Illustrator exclusively. If you're making 3D graphics, you can opt for programs like Cinema4D or 3DMax. You can pull a Sean and use Maya, which is awesome because I have never seen that program used for print design.
Heck you could whip out Flash and make your cards there. It all depends on the style you want to have.
I suggest avoiding free programs because even if they can accomplish what you need, you will never learn the skills you need to move into a professional program. That being said, I know very few people that actually pay for the high-end programs. If you have insane morals, just keep re-downloading the 30-day trials while you learn.
You won't be good at anything until you use the programs for at least on year. You won't be ready to make an amazing deck for at least three years. You won't master it for at least five, and trust me you will never be the best at it. Heck, if you take a break for any amount of time, you will be so out of date you practically have to relearn everything. I stopped designing for a year and a half and I came back to techniques I had never even heard of.
When we launch our design board, I will be providing resources for people to get a jump start on their art and learn the trade as a hobby.