You should have no problem entering the serial number for Lightroom 5 into the new computer. If you bought an upgrade from Lightroom 3, you will need to also provide the serial number from Lightroom 3.
So when I get the new computer going, I am assuming I download the full version, trial version, or the free trial version or are all the same? This part Im lost with.
There is only one version.
Download it, enter your serial number(s) and you're ready to go.
You are awesome, thank you very much...