You can get a "domain name" for your new t-shirt website anywhere. The domain name is separate from the actual hosting and website design.
Places like godaddy.com, mydomain.com, even yahoo can sell you a domain name at a fair price.
That would be totally separate than paying for the hosting of your webpages and the hosting would be totally separate from paying for the "design" of your web pages.
There are literally 1000's of web designers and web hosts out there. Here's a good article that should help you narrow down the right choice for you when looking for a web designer:
http://www.sitepoint.com/article/how...a-web-designer
Here's a thread that talks about places to find t-shirt designers. Many of those places also have web designers for hire:
looking for graphic designer for caricatures/cartoon designs
Here's a thread where folks shared some of their favorite webhosts:
Good Web Hosts?