As Drano explained, It's very simple to have your new broker handle the transfer of all the shares and other financial instruments in your account at the old broker. EXCEPTIONS would be when anything is in a margin account and you have outstanding borrowing against your short positions. Short positions would have to be covered and closed out*. Not sure if any options you've written will need to first be repurchased and closed out.

*My experience with short positions and options dates way back to the 1970's, so some things might have changed. Ask your prospective new broker how they do this including the "paperwork" and how long the process takes.