If you can see the serial number from the splash page, do this to get it in a copyable form:
Open TMG and use the Help > Technical Support command. On the screen that opens, click the Trouble Report button at the bottom left. Let that report open, and you should see your name, email address, and serial number near the top of the page.
Using that information exactly, even if some of it has since changed, enter them as described in my article on Moving Data to New Computer or Version.
Once you do that, make sure you save that information in a safe place where is cannot be lost by a computer failure. You also need to save a copy of the installer in case you need to re-install in the future. See my article on The Future for TMG Users for how to do that.