I don't know about how people go about this, but I do know it's very hard to get into the US without a work permit visa and they're hard to get without an acceptable reference from someone who lives there or is employing you.. :/
If you are under 30 then you can apply for an oz working visa for one or two year, it was two when I was there, this means you can live and work freely for two years.
America is a very diffrent matter they do not do working visas of any kind to UK citizen (green cards etc anything like that is only if you live in a developing country, eg. Mexico etc)
You cannot work in America unless you find a company that is willing to sponser you and they can prove that only you could do the job and not a US citizen. this as you can imagine is difficult!. If you really wanted to work in the US you could find a Uk company that has an office in the US so you would tecnically be employed in the Uk.
Internship etcs in US are not ushually paid and most would expect you to work in the evenings to cover your own costs, places like camp America and Buntac (excuse spelling) provide UK young citizens with places in kids summer camps etc but I have never seen anything like this for graphic design
Failing that you could always marry an American but even this isn't easy I have friends in California and it was a nightmare for them sooo much paper work!