Hello, Meemster_1344.
Follow the directions found in this support document to create a full Time Machine backup of your old machine onto your new LaCie drive.
Back up your Mac with Time Machine – Apple Support:
https://support.apple.com/en-us/HT201250
When that is complete (could take several hours depending on how much data), use the new backup drive to eventually migrate your data to your new Mac when you first turn it on.
The Best Way to Migrate Data from an Old Mac to a New Mac:
When you’re ready to setup your new Mac the easiest and most reliable way to migrate your data and files from the old Mac is simply to connect the old Mac's Time Machine external backup drive to the new Mac using a USB cable before you turn on the new Mac for the first time. You may need a USB/USB-C adapter for this; easily found at many resellers for ~$10 or less.
When you start your new Mac for the first time it will launch Setup Assistant and lead you through the setup process. Eventually Setup Assistant will ask if you are migrating from another computer. You are, so select that option and follow the on-screen prompts to completion.
If you have already run and completed the Setup Assistant on your new Mac, then you can use the Migration Assistant application to move your data over. Connect the Time Machine external backup drive using a USB cable and launch the Migration Assistant app (located at Applications > Utilities > Migration Assistant). Then follow the on-screen prompts to complete the data transfer.
When migrating from an older macOS to Catalina or Big Sur, I suggest only migrating the User Account(s) and nothing more. Do not migrate Settings and especially not Applications as High Sierra and earlier will have outdated apps that won’t work on Big Sur.