I bought the product and am trying to install the key on the trial version - please tell me how to do it
If you downloaded the trial version from the creative cloud offering you cannot activate it with a serial number. It is licensed differently and doesn't use a serial number. It will have to be uninstalled.
If you downloaded the standalone Lightroom, on the trial screen that first appears there is an option to indicate that you have a serial number. Choose that option, and you can enter the serial number you have received.
In the event that you are using the creative cloud version, here are the Windows and Mac links to the standalone trial version.
If you find that it's necessary for you to uninstall/reinstall as instructed, don't worry about losing your work. Your catalog and all your images will still be intact after you install the right version.