Is it common practice for a new employer to garnish your first week's salary until you resign?

No one can "garnish" any wages without a court order for a debt.

It is common practice for working places to have pay weeks and for you to be paid for the work on one week, at the end of the next, this is to allow them to have time to do payroll, print checks and so on.

For example tomorrow is 3/7/08 so the work I did all this week will be paid on 3/14/08. This is not being held back, it is not being garnished, it is merely being processed. So in general then you do get a check for the week after you latter quit, but it is not your first weeks check, it is still your last check,

There is merely a processing time to receive paychecks.