If l purchase a car today and they sign the title over to me. How many days or months do l have before l am legally obligated to transfer the title into my name?

Do not know about Virginia, but in Indiana, you have 30 days from the sale date that was put on the title, otherwise you would be assessed a late fee.

That is why most seller just sign the title and the buyer waits to fill out the rest of the information when they are in the DMW parking lot. ;-)

The site below tells you everything you need to know about titling a vehicle in Virginia including fees. There was too much to cut and paste here.