It seems the general trend would be something in the like of getting a visa, start looking for a job and be lucky your employer can get you a green card. After this 5 years of living in the states before you can become an American citizen.
Please do correct me if I'm wrong!
The most important factor here is your preference category when applying for a green card. There are 3 buckets, EB-1, EB-2, and EB-3, in decreasing order of desirability. Which bucket you fall into is a combination of your degree(s) and your relevant work experience.
Each bucket has a longer wait than the previous.
The second factor is your country of origin - the system is quota'ed based on it, so the more applicants coming from your country of origin the longer the line is. For the most part this is a minor factor except if you're from China or India, in which case you're pretty fucked.
I believe (though I'm not sure) there are also other ways to expedite processing for $$$, but this mostly surrounds various form filings that can otherwise take months to process. The actual line itself cannot be shortened.
Yet another alternative is to start your own company in US and invest about half a million $ (officially, unofficially have seen it done for 100k).
Yet another alternative would be to get a job in academia that has H1B uncapped.