Once you have a Visa you can pretty easily get a Social Security Number by visiting the Social Security Office. To learn more about social security and how to find a Social Security office, go to the Social Security Administration web site.
They are a bit pedantic and you can easily get rejected for having the wrong paperwork. They take many weeks to process and they often lose the applications which can make for a frustrating experience when you're waiting on your SSN in order to open a bank account, apply for health, etc.
It is possible however to open a bank account without an SSN. If you go to a branch and speak with a representative, some will be willing to open one based on showing them a work visa and a promise to fill the SSN in once you get it.
My recommendation is to treat your SSN as your number 1 priority after you get your Visa approved.
If your partner is travelling with you on your Visa class you should also make sure that they apply for an Employment Authorization Document (EAD) which will enable them to get a SSN so that they can work.
Not having a SSN in the US can be a real pain so get moving on it quickly. [I recommend waiting for a couple of weeks after you arrive - otherwise you can be flagged by homeland security leading to a couple of months delay on the SSN - which hurts. MH]