Make sure that you are listed as the registrant (with a correct email address), and the domain is unlocked with your current domain registrar.
Then, initiate the transfer from your new registrar. You'll recieve a confirmation email (sent to your registrant contact). After that, the losing registrar (your old registrar) has 6 days to approve it.
Network solutions is a blood sucker, almost no one charges for failed transfers anymore.
To transfer to someone else, its very simple. Simply give him/her the login name/password after you have recieved your compensation for the domain and have the person change the registrant contact to his name.