What most people refer to as "sponsorship" usually means the infamous H1B visa. These are getting harder and harder to qualify for, as there is a ridiculously low cap on the number available. There are now many more applicants for the H1B visas than there are visas available and they have gone to a lottery system - so even if an employer is willing to support your visa application by hiring you, you are still subject to the luck of the draw.

To find a job in the US you need to have some skill or experience that is not generally available in the job market over there - it can be language ability or experience in the international marketplace, for example. But it's not easy to get a US employer interested in hiring someone from overseas - it's expensive for the employer and it's quite a bit of hassle to justify taking a foreigner over someone locally or someone who already has the right to remain in the US.

How soon you can apply for US citizenship depends to a certain extent on what sort of visa you are on. Some visas can't be converted to green cards (the first step toward permanent residence) and others can be converted fairly shortly after your arrival.

One of the best ways to go to the US is to work for an international company in your home country for a time (at least a year) and then get transfered to the US branch on an L category visa. This is often limited to manager level personnel, and each company has its own internal policies about transfers.
Cheers,
Bev