This has become fraught with peril lately, even after initiation it can be far from easy to figure out what to do next. What to do next, after the transfer has initiated, is, we browse here:
www.networksolutions.com/manage-it/transfer-status.jsp
and we put the transfer authorization code, from the outgoing registrar, after finding the appropriate Network Solutions account number for the order, and scrolling in the above page (if there are more than one; if you do a lot of this, there will be a pile, if not, just one) to find that account number. Then open the account and enter the code.
If you have trouble finding that account number, browse here:
www.networksolutions.com/my-account/order-history
and the account number will be listed with the order for the transfer.