Updated: May. 2020Easiest way to copy or transfer Sims 3 game files, (like saved games, mods, screenshots, etc.) is to save the whole Sims 3 folder to a usb key or external hard drive. - This way everything is saved and backed up. Plus it's easier to copy the whole folder at once.
How to Copy -
Right-click on the Sims 3 folder, select Copy and Paste it to your USB key.
If you want to break it down even further in the way of saving just saved games. Then just copy the Saves folder to the USB key or external hard drive.
More information can be found in the post linked below:
How To: Saving folders/files
http://www.carls-sims-4-guide.com/forum/index.php?topic=16414.msg313134#msg313134Original post in spoiler tag below:
I recently done this from my laptop to my new desktop. I just copied the files I wanted onto a disc and then transferred it to my desktop from doing copy and paste from the disc. I don't know much about computers but it worked out very well. Someone might come along with better methods though. But this way worked out for me. I hope it helps you in some way.