In terms of artwork I prefer Photoshop because I'm familiar with it. But Corel or Illustrator are also great options. Go with what you know how to use. If you're unfamiliar I'm sure you can get the gist of it by watching youtube videos and tutorials. If you really want to learn the programs maybe join a class at a local community college, they usually have classes to learn about design software. As for equipment, what's your budget? There are a bunch of packages on this website:
How To Start A Screen Printing Business, Silk Screening Starter Kit, , Silk Screen Printing Starter Kit
The ones that are in the $7k and up are more production friendly and they are good for starting out also, but they give you some growing room.
The PRO 2 Complete Professional Screen Operation HV Package
That package is the one I recommend most because it comes with a decent size dryer. I know it's the most expensive package, but you can get a manageable monthly payment by leasing with them. My Equipment was about $12k and I leased it when I was only 18 or 19 and I've been making the payments no problem.
It all depends on what you wanna do. If this is more hobby related or you just want to start out really small then you don't need all that stuff. The cheaper packages will get you printing, however it's a more time consuming process.