Download Elements to the new computer from this link. Download Photoshop Elements | 14, 13, 12, 11, 10
When you download it. Save it for future use. Copy it to a disk, flash drive or any place that works for you. That way you can avoid downloading it in the future.
What program or programs do you have? Just Photoshop Elements 14, just Premiere Elements 14, or both?
Although the link already given shows a Photoshop Elements screenshot, the link for Premiere Elements is to the right
in that page
Your thread was posted in the Adobe Premiere Elements Forum where you can get troubleshooting help with Premiere Elements.
For Photoshop Elements, please take advantage of the specialized expertise in the Adobe Photoshop Elements Forum.
Just in case note: Be sure to put your purchased serial number in safe accessible place or places.
Add On to my post 2....
From what you wrote, you already have the installation files for Photoshop Elements 14 or Premiere Elements 14 (whatever you have). You need not download additional installation files for a backup. Use what you have, copy/transfer them to a USB flash drive (for long time storage and for immediate use) for transfer to the other computer.
As a result of the purchase, do you have an Adobe Premiere Elements 14 Folder on your desktop or saved some place else? Put that Folder on the USB flash drive and install the program on the other computer, using the set-up.exe inside the PRE14 Folder in the Adobe Premiere Elements 14 Folder.
Remember, Adobe allows you to use the same purchased activated serial number on no more of 2 of your computer, and then use only one at a time. Also, I am assuming that you are using comparable computer operating systems in this. No Windows to Mac or Mac to Windows and the like.
Consider and determine what works for you.