I am getting a new computer, can I transfer my lightroom 4 from my old one to my new one ? If so how?
Download Lightroom 4.4.1 from this link, install, then enter your serial number.
To move your catalog and photos, see How do I move Lightroom to a new computer? | The Lightroom Queen
I know this may sound daft, but how without trawling my mailbox do I find the serial number?
You ought to have a way to keep track of these important things that you paid money for, and furthermore, every mail application I have ever seen has a search feature ... which should find the answer for you in a few seconds
Nevertheless, maybe this will help: Find your serial number quickly