Usually all you have to do is provide the plan administrator at your new employer with the account information for your 401(k). They'll do all the paperwork.
This is called a 401K rollover. You should receive 401K rollover forms from the new employer as part of joining their 401K. Or contact the 401K company for the form.
Contact the 401k plan trustee of your old plan and inform them you want to transfer your balance to a new 401K plan. They will send you a form that you will need to fill out and return. Keep a copy for your records.
