All you need to do is:
1. Copy your Sims 4 folder to a flash drive or other backup media of choice.
C:\Users\<Username>\Documents\Electronic Arts\The Sims 4
2. Copy that folder from backup, to the Electronic Arts folder on your new computer after Sims 4 is installed.
Read more details in the link below:
How-To: Copy Sims 4 game saves to another computer
http://www.carls-sims-4-guide.com/forum/index.php?topic=21899.msg449653#msg449653
Original post under the spoiler tag.
I got the Sims 4 in December and since then have purchased many expansions and built a large family through generations. I recently got a new computer and wanted to transfer all of the data from old to new. I have spent hours on this website and various others, trying to make it work. I've copied saves folders and tray folders and reinstalled everything. Nothing is working. I'm desperate, my old laptop is about to give out for good and I'll never have the data.