Put the 2011 laptop into Target Disk Mode to see which symbols are displayed on the laptop's screen. These symbols identify the protocols you can use for the transfer.
If you see the Thunderbolt symbol, then you should be able to use the Apple USB-C to Thunderbolt Adapter.
If you see a Firewire symbol, then you can use the above adapter along with the Apple Thunderbolt to Firewire Adapter (maye there is a third party USB-C to Firewire Adapter).
If you see a USB symbol, then you can use an USB-C to USB-A cable.
Or you can remove the hard drive/SSD from the 2011 Mac and use a USB to SATA Adapter, drive dock, or enclosure to connect it to your new Mac. Just make sure the device is compatible with an M1 Mac. Some USB devices contain controllers that are not compatible with an M1 Mac. I know the JMicron USB controller is known to have issues with an M1 Mac. I don't know if ones have known issues. Make sure whatever adapter, drive dock, or enclosure you purchase is compatible with an M1 Mac. I have used a Tripplite USB-A to SATA Adapter with an M1 Mac. Here is a USB-C version (I haven't used the USB-C one yet):
https://www.cdwg.com/product/tripp-lite-6in-usb-c-gen-1-to-sata-iii-adapter-w-uasp-2.5-hard-drives/4426781?pfm=srh
If your old Mac is sluggish, then either you have a software issue, a file system issue, or you have a failing drive. If you have a failing drive, then I hope you have a good backup since it may be difficult to retrieve data from a failing drive (the more you use a failing drive the worse the failure will get where even a professional data recovery service won't be able to retrieve any data.
You should always have frequent & regular backups. This is even more important for your new 2020 laptop because it is impossible to recover accidentally deleted data from an SSD plus an SSD can fail at any time without any warning signs. If the new laptop has a hardware failure, then an expensive professional data recovery service would be required to attempt to recover data from the broken Mac.
