Can I move to ny from nj to start a new life with my wife?
How do I go about a transfer of probation?

Only if your PO allows it (depending on the specifics of your case... they probably will)... But you better get permission in writing to transfer it (its gets a bit more complicated than just a letter) BEFORE you go anywhere as it might take some time.

Start a new life ? We can improve the one we have, but our past is always there.

You start by visiting with PO and making a written application for a transfer. They may want to know details, so do you have work arranged, a place to live ? It is doubtful that they will allow a transfer unless you have proof that you will have a reason to move