How do I safely replace the SSD that has my OS on it?

Cloning is probably the most direct route. Assuming Windows, put the new SSD in a compatible USB enclosure, then use drive cloning software to copy the entire 1TB system disk to it. Copy all the system and recovery partitions, not just the main Windows partition.

Back up anything you can't afford to lose first. Check the source and destination carefully in the cloning tool because the destination will be overwritten. Once it finishes, shut down, swap the drives, and check that Windows boots from the new SSD before erasing or repurposing the old one.

The clone may leave some of the new drive unallocated. Depending on the partition layout and the cloning tool, you may be able to expand the Windows partition during cloning or afterward.

You can avoid buying an enclosure if you're comfortable opening the PC. Shut it down, temporarily remove the game SSD, and put the new SSD in that slot. Clone the whole OS disk to the new one, then shut down and replace the old OS disk with the clone.

Boot and check the new drive before reconnecting the game SSD. If the machine tries to boot from the wrong drive, check the boot order in firmware settings. Keep the old OS drive untouched until you're happy everything works.

A fresh OS install is another option if you don't mind setting things up again. Back up your files, install the OS on the new SSD, then restore your data and reinstall your applications. It takes more work than cloning, but you aren't carrying the old installation over.

If the hardware swap or clone feels too stressful, a local PC shop can do it for you. Ask for a price first and make sure your important files are backed up separately either way.