How can I get my laptop's existing system onto a new replacement drive?
A few ways to approach....


Clone your existing drive to a drive image on the 1TB (if it has room). As @revolver mentioned, Carbon Copy Cloner is a good tool. Super Duper does very much the same thing. Your choice. Swap drives and then restore. You'll need a boot drive that has your clone tool. The 1TB could do this, or a flash drive, etc. 
Plug the new drive into the USB/SATA adapter. Install OS X on the new drive so you get the recovery partition setup properly. Then clone directly from internal to external drive. No drive image involved. You can even reboot while in this config to confirm everything worked. Then swap the hardware. 
Run a Time Machine backup. Install the new drive. Install OS X, with its recovery partition. Boot to the recovery partition and restore from Time Machine.
Install OS X on the new drive and use the Migration Assistant tool to move all your data and apps over from Time Machine backup. This has the advantages of removing some cruft and getting you a "fresher" environment.


I'd pick #2 as you can test the new drive before you commit to the hardware install.