It all depends on the OS running on the old MBAir and the exact model of the newer laptop.
It may be easier and safer to perform a clean install of macOS on the newer Mac, then migrating your stuff over. Keep in mind that the newer OS may not be able to run some of your older apps without those apps first being updated. If you run macOS 10.15 Catalina on the newer Mac, then you need to know that any 32 bit apps on the old MBAir will not work unless you can update them to 64 bit versions. There are some limitations to using Migration/Setup Assistant for migrating apps since it doesn't transfer the software license for some third party proprietary software like MS Office. In fact it is unlikely older versions MS Office will even work on the newer OS without purchasing the newest version.
32-bit app compatibility with macOS High Sierra 10.13.4 and later - Apple Support
If you choose to clone the drive, then Carbon Copy Cloner is a good choice as it is very easy to use and I've never had any problems with it.
I'm assuming the newer Mac is a non-Retina model since you mention a memory upgrade. Macs are very picky about the memory they use so make sure to only purchase memory from Crucial or OWC. Make sure to use the tools on their respective websites to get the exact part numbers that are compatible with your Mac.
Make sure to install an SSD as that will be a good performance boost, but stay away from the cheap low end models as they are as slow as a hard drive. The Crucial MX500 series SSD is a good choice as it is a good compromise on price & performance. Stay away from the Crucial BX500 series as it is a low end budget economy model that is as slow as a hard drive and tends to overheat easily. Many SSDs today are unfortunately low end models so be very careful. It is also extremely difficult to distinguish between the good SSDs and low end models these days as the manufacturers are trying to trick consumers. Don't rely on reviews from even one or two years ago as some of these SSDs may have changed without a corresponding change in their model numbers. Yes, the SSD manufacturers are that sneaky!
If the newer Mac is a Retina model, then you cannot upgrade the RAM and you need to know that the Retina laptops use a proprietary Apple PCIe SSD connector. If the laptop never had macOS 10.13+ installed, then you will not be able to use an NVMe SSD.
You need to make sure that the newer Mac does not have a firmware password lock enabled. You can do this by holding down the Option key just after the startup chime. I highly recommend installing macOS on the newer Mac to see if you get any notice about the newer laptop being managed. If so, then return the laptop for a refund unless you can get the previous owner to remove the firmware lock or have the laptop removed from the management system of the school or business. I would perform another clean install of macOS (erase the whole physical drive) to make sure you are not presented with a message about the laptop being managed.
Another benefit of installing macOS is it should update the system firmware on the off chance the system firmware never got upgraded properly by the previous owner.
