the most time-saving solution is to use the command "dd" to clone your disk to another disk. Of course, the storage size of the newer disk should be equal or larger than your current disk storage.
sudo dd if=/dev/sdX of=/dev/sdY
WARNING: THIS COMMAND MAY DESTROY YOUR DATA. PLEASE MAKE SURE WHAT YOU ARE GOING TO DO BEFORE USING THIS COMMAND.
The "X" of sdX and "Y" of sdY depend on your machine.
sdX means your current disk label.
sdY means your target disk you want to move in.
Please use "mount" or try to google to know your sdX label of your system. I could not judge the exact lable name for you because this varies between systems. In most cases, your current disk is /dev/sda and your target disk will be /dev/sdb.
use this command to get more information in your terminal:
man dd
You do not worry about driver issues too much. Linux kernel will detect and (try to) enable hardware components during every booting.