What version of software are you running: CC, CS6, CS5, CS2, what?
The general process would be to install the software on the new computer and run in trial mode for up to 30 days but make your transition period quicker than that in case something happens.
Do whatever you need to do on the old computer including copying preset files and other things across to the new computer so you're satisfied the new computer is working.
Deactivate the software on the old computer, by either:
a) For CC deactivation is as simple as signing out of the CC Desktop app on the old computer and never signing in, again.
b) For pre-CC you can use Help / Deactivate on the old computer to deactivate the software.
Then activate your trial on the new computer using the serial number(s) you had on the old computer if pre-CC or by signing into the CC Desktop app on the new computer.
For pre-CC you will need your serial number(s) associated with the software so make sure you have those before you start. You would need at least the current version of software serial numbers, and if that is an upgrade, then the previous version of software serial numbers.
if there is something special about this installation like it is an educational version or Adobe had to hack something on their server due to some special circumstances to activate one of the older computers, then activating the new computer may not be possible without those special efforts being repeated.
Here is a variation of the instructions I have that says to deactivate on the old computer before installing on the new computer. I would think you'd not want to do this until you were ready to start working on the new computer, though.
Earlier deactivation may be necessary if you want to use some sort of Apple migration tool to move the settings across, so here is what they say for CS5:
The licensing isn't going to be migrated properly by any Apple product so even if you want to move things across you need to do a fresh install of the programs, themselves, on the new computer so they are licensed properly.