Format your new drive from an external connection (enclosure) with disk utility using MacOS journaled. It's called "erase" in Disk Utility, but it will format the drive. Then, clone your original drive to the new drive using Carbon Copy Clone or SuperDuper. When the drive is cloned, it should be bootable, and you can test it by actually booting from it before installing it. Once you're sure it works, swap out the drives. If you named the drive the same exact name as the original drive, Time Machine should just keep on backing up the drive as if it were the original, with no problems.
That's just one way, but you get the exact same 'drive' you already had, including all installed apps, prefs, etc... only with more space/speed depending on what your new drive is.
You could just do a clean install of OSX to the new drive, and them use Time Machine to restore your files... but it's not a clone. It all depends on what you want.
I used the 'cloning' method when I replaced my original drive, and it worked perfectly. Some folks have their own way of doing it, so there's no single 'correct' way of doing it. Good luck with it.