Summarize the following:
Open the dialer on your phone and type *#06#. A screen will appear showing the code. Make note of this number for later use.  Make sure to write the code down, because it cannot be copied to your phone's clipboard. You can also find the IMEI number printed on a sticker underneath the battery of the S3. However is not recommended to take it from the box or underneath the phone's battery as it might be different and your unlock code will not work. There are multiple sites online that offer to unlock your phone for a fee. Read reviews and make sure that other customers have been satisfied with the service. You will need to provide your phone's IMEI number. After finding a trustworthy site, order your code. Make sure to enter in your carrier, model, and IMEI number. Do not trust sites that claim to unlock phone's for free. If they prompt you to fill out surveys or participate in affiliate programs to get your code, it's most likely a scam. Depending on the service, this could take a few hours or a few days. You may receive your code via text, email, or phone call, depending on the policies of the site you pick. Make sure the card is not from your old carrier. When you are prompted to enter your unlock code, enter the code your received from the website. Verify that you have service and that the code worked correctly.

Summary:
Retrieve your S3's IMEI number. Find a paid unlock service. Order a code. Wait for your new unlock code to arrive. Insert your new SIM card.