It is currently running version 14 long term support IIRC. What is the best path forward to bring my operating system up to date? Will I be able to jump straight to the latest long term support package? What pitfalls do I need to avoid?
That is very old. Backup anything important to a safe place (i.e., not that computer), and reinstall a supported version of Ubuntu. If the old version won't boot when you make a backup, create a live CD with Ubuntu 20.04 LTS, and use that to backup your files (remember, live CDs reset when you reboot, so don't copy your files to the live CD*). For the newly-installed version, if you like LTS, you would use 20.04, but if you want the latest stable one, get 21.10.
*Technically, they don't have to, if you setup persistence. But it is easier (and safer) to just copy the files to somewhere safe, like an external hard drive.
I need to get back on the internet with some sort of a router/repeater. I have an inhouse router upstairs I use to access the web with from other devices. I need something downstairs that will allow me to access the web through an ethernet port on the Linux Desktop. So whatever device I buy needs to have an ethernet port.
I would be surprised if your desktop, despite its age, doesn't have an Ethernet port. Just plug the ethernet cable into the desktop during system installation, and it should autoconfigure it.
If it really doesn't have an ethernet port, you can get a PCI-E Ethernet card for not very much money on Amazon. TP-Link is a good brand (no affiliation to it, other than using it). If your device is old enough to have PCI, not PCI-E, you can get PCI Ethernet cards, too (StarTech is also a good brand). This photo illustrates the difference between PCI and PCI-E. If you prefer USB, there are many adapters for that, too.