How much internal storage did you have on the old Mac and how much was stored on it? The largest internal storage was a 3TB HD used on some iMacs, so you don't have more than that. If you do own a 3TB iMac and most of that is filled yes, you are correct it will not fit on a 2TB Mac. So the solution is to move large libraries which are typically:
libraries to an external HD first on the old Mac. Apple explains how this is done in Move your Photos library to save space on your Mac, the same process is used for Music and Movies. You will need a high quality External Hard Disk such as the OWC Mercury Elite Pro or an external SSD. Once you have successfully moved the libraries to the External Hard Disk then everything becomes pretty simple. To teach the apps that use those libraries (Photos app, Music App, TV App) where the libraries now reside (on the External Hard Disk) simply double click the libraries in their new locations and that will tell the apps where the libraries are now located. Once you have done that step with each library then it's simple, erase the old libraries from their original locations and empty the Trash.
Next, simply execute Setup Assistant on the new Mac by following the instructions in Setup Assistant and Migration Assistant
