The built-in way to do that is on FIRST run with Setup Assistant, or later with Migration Assistant.
Their selector window looks like this one:

this is the Apple article on the subject:
How to move your content to a new Mac - Apple Support
The article includes references to Target Disk Mode, which may be possible if you have the right ports available.
if you can get both disks mounted at the same time, you can simply open TWO finder windows -- one of the old drive, one of the new drive, place them side by side on your screen, and drag and drop whatever you would like to copy across. [Once one Finder Window is open, just choose New Finder Window from its File menu to open another, the navigate to wherever you like.]
The downside of manually copying is that there may be quite a few settings files that would be left behind, as they are either hidden files or in hidden Directories, like the User's Library folder, ~/Library.