If you have an M1 Mac, then it is possible the external drive is not compatible. There have been numerous posts on these forums where people have had issues using some external hard drive with an M1 Mac. There are not a lot of details or good information on this issue.
Make sure to connect the drive directly to the Mac and if you must use an adapter try using the Apple USB-C to USB-A Adapter as some other adapters, docks, or hubs may not always be fully compatible with a Mac or all peripherals.
As for updating your old laptop you may first need to upgrade its OS to macOS 10.11 El Capitan before being able to upgrade the laptop to macOS 10.13 High Sierra. You can find a link to the macOS 10.11 installer within this Apple article which also includes instructions for creating a bootable macOS USB installer:
https://support.apple.com/en-us/HT201372
Make sure to read the instructions carefully even if you don't want to create a macOS USB installer because there are several steps needed to properly extract the macOS 10.11 installer from the downloaded .dmg file. I highly recommend creating a macOS USB installer as it gives you more options in case something goes wrong.
Before upgrading macOS I highly recommend checking the health of the hard drive on the old laptop since upgrading macOS may put a strain on such an old drive and reveal that the drive is failing. You can use DriveDx to check the health of the hard drive. Post the report here using the "Additional Text" icon which looks like a piece of paper.
You can also try removing the old hard drive and connecting it to the new Mac using a USB to SATA Adapter, drive dock, or enclosure. Here is one that should be good although I've never used it (I have used other Plugable products and liked them):
https://plugable.com/products/usbc-sata24
Here is a standard USB-A adapter which I have used, but it will require a USB-C to USB-A adapter to use with the latest Macs:
https://www.tripplite.com/usb-3-0-superspeed-to-sata-iii-adapter-for-2-5-in-to-3-5-in-sata-hard-drives~U338000SATA