You will find that most brokerage houses and banks who offer brokerage services will be helpful to you in facilitating the transfer. There is (or was) no charge to complete this process at one of the largest banks in the US. If you bank with an institution that has a broker or affiliate broker, start there.
You need just a few things to make the transfer happen.
If you have possession of the certificate (which it sounded to me like you do) you will of course need this. Do not sign the back of the certificate until you are with the person completing the transfer for you!
You need to write a quick letter of instruction, specifying how you want your shares transferred. ("Transferred" is the terminology used when changing names as well as changing ownership).
List the name (your new name) and address as the "new owner". Include your social security or tax payer identification number which is used to report dividends to the IRS. Also if you are adding any owners, you would identify the preferred form of ownership (i.e. Joint Tenants, Tenants in common, etc.) In this case you will state the stocks are to be held in your name only.
The transfer must then be signature guaranteed by a financial instituion participating in the Medallion Signature Guarantee Program. (Most brokers/banks and financial institutions participate). When you are sitting in front of the person who will guarantee your signature, you will sign the certificate exactly as your name appears on the current certificate. There is a stock power form on the reverse of the certificate that is filled out at the same time.
These forms (the certificate and the letter of instruction) are then mailed to a transfering bank in New York. There is occasionally a charge to mail, which is the fee to certify the package as well as 2% of the value of the certificates (insurance). You might want to know the value of your certficate in advance and request the insurance. 2% is about what it costs to have new certificates issued if these are lost in the mail.
The bank or broker will mail the transfer package for you typically and you will receive your new certificate in the mail in two to six weeks.
Now as an advisory service

If you have any other stocks or mutual funds, you might consider adding this certificate to the safekeeping that those funds are under. The value of the certificate will be added to your other investments. If you do not currently have other investments, hang onto this one, but if you ever do start purchasing any investments, have this transfered to that account under safekeeping. This eliminates all of the issues of handling a transfer yourself. It simply takes a letter to the brokerage firm and verification of the ownership to complete processes like this...as well as allow for sales and dividends to be processed quickly and efficiently. For a small account there would be no charge for this. It's the value a broker would provide