How do I become an resident in the Bahamas or other Caribbean tax haven ?

I am currently a EU citizen but not from the UK, I am not rich but are
working on it, I am spending a lot of time in the US but do not want
to reside there.

Is there a simple process to go through if so how does it work and how
long time does it take ?

Thank you for taking time to answer my question !

For Bahamas, you will find your answers in the "Employment and
Residence" section at:
http://www.lowtax.net/lowtax/html/jbaoltr.html#employment

For other Caribbean countries, see:
http://www.lowtax.net/lowtax/html/jbahom.html , select the country of
interest, then click on the "Employment and Residence" link for
details on that country.