I am from India and had I been in your place, I would opt for domain names like these: - designerindia.com - indiandesigner.com - graphicdesignerindia.com - indiangraphicdesigner.com - webdesignerindia.com - indianwebdesigner.com You should replace the word "india" or "indian" with what is applicable depending on your country. Such domain names have the potential of ranking well in Google and other search engines. This means targeted traffic and therefore, more business! Best of luck with your new venture, my friend!
you can lookup for names that contain your work subject, but it would be nice to get something memorable, catchy and easy to spell too.
you can use most domain registration sites, like www.DirectDot.com i sometimes use domaintool.com to get some ideas for some names. and then register them with my own domain registration service. check my sig for details