I am picking up a newly purchased computer tomorrow and want my files transferred to my new computer. What is the best way to back up - iCloud?

iCloud is not a backup tool. Use Time Machine to back up a computer.

And once you have that Time Machine backup of your current Mac, you can connect that to your new Mac before you power it up and proceed with the initial setup steps. Among these it will ask if you want to migrate files, and a Time Machine drive is one of those options. It will search for a mounted Time Machine drive, you click on that selection, and it will proceed to migrate your files and settings (including email, etc) to the new Mac. Be patient.