What timestamps does rclone look at?

Is it just the date file last modified (I hope so), or is it the date file created as well? I keep the files on the external 1 Tb HDD, synced to Google storage, but someday I will have to move these files to 2 Tb HDD. So, when I will do rclone sync from the new HDD for the first time, will it transfer ALL the files, because dates of creation are different? Or is there some switch turning creation date on/off?

Yes

When you copy the files from old HDD to new HDD just make sure you preserve the modification times and all will be well (eg cp -a on linux).

You can use rclone to do this copy and it will definitely preserve the modification times.