What process needs to be followed in order to acquire an American Green Card and subsequent citizenship if one is in the US on an H1B ?
There are many paths to both a Green Card and Citizenship, including employment based, family based (e.g. marriage) and various special interests.
Since you are in the US on an H1B, you are likely employed, and if you are lucky your employer will sponsor you for a Green Card. It is a long and fairly complicated process. You should contact your HR department; however if they are not used to doing this it may be an uphill struggle to get them to spend the time and money on an application. This slide show from "Coming to America" (slides 8 to 15) has a good summary.
As for citizenship, most paths to that go through the Green Card. Once you have a Green Card there is a waiting period before you can apply. There are other routes like joining the US Military. See the USCIS website.