If you're moving the chains after a play, you have a new series of downs...
You will ALWAYS start a NEW series of downs 1st and 10.
If you are NOT moving the chains after penalty enforcement, this 1st down is a repeat of the previous 1st down, and NOT a new series. In these cases, you can have a 1st and 1, a 1st and 10, a 1st and 30.... Whatever it ends up being.