Many people ask me which graphic art software I use. I actually use Illustrator, Corel Draw and sometimes Photoshop. However, I would suggest Corel Draw for anyone starting out because it is a bit easier to learn and user friendly. The Corel package has all of the same functions as Photoshop and Illustrator, although Photoshop may handle photo retouching a little better. And it has a moderate learning curve. You will still need to learn a lot about the program to successfully do what you want but in general it is easier to learn than Illustrator.
There are many tutorial videos about creating art for screen printing on YouTube. You may want to look at a variety of tutorial videos on Illustrator, Photoshop and Corel Draw to help make your decision. There are quite a few of these. You can start with my favorites list and check out the artwork tutorials there and then look at the related videos to find more. Or you can do a YouTube search for them too.
The bottom line is you will need graphics software that handles vector art. The generic file format for vector art will be EPS which stands for encapsulated post script. This is the file format that you would request from your clients. I will most likely write an article for the website about this sort of thing in the near future.