For what you are talking about, you don't HAVE to have a new computer. They can always send you jpegs or PDFs of the work, which your computer should be able to view without any issues. That being said, if you need a new one, I'd try and get one that is as powerful as your budget would allow.I am not a printer or embroiderer, I will hopefully be launching a line of fitness wear all done via my website. Even though i was a design major many years ago, i will be using a designer but I'm sure that we will be exchanging files and such. Currently, I just have my crappy laptop. Do I need specific hardware/software to have an e-commerce site and to exchange ideas and info with my designer? I think I should purchase a new computer for the business.
As for software, that is a hard one. There are two big contenders out there - Corel and Adobe, and there are a lot of opinions on which to get. I would only look into these if you are going to be editing files. Of course, it would be easier for you to look at the same software your designer is using.
As for what I use, Macs at home and work, as well as PCs.