Remote Desktop Best Options?

Setup a Guacamole server for them. It can do RDP, VNC and SSH. Really easy to use, no client/host machine software needed (at least not for RDP/ssh), never used it with VNC. Also, really old Windows machines, not sure how they would work with RDP but I imagine they would.

On your end, you just go to the Guacamole site, login and pick the connection you have setup for a machine and it connects via the browser, no extension or plugins needed.

Personally I avoid Teamviewer like the plague. It constantly has security issues, which are usually discovered after its a problem. I wont even install it on my own machines unless its inside a VM, and even then its only to connect to others to help them.

If Guacamole isnt an option...

If your on a Linux box connecting to all these I would recommend Remmina (https://remmina.org/) for RDP, closest thing I have found to the native RDP app on Windows. SSH shouldnt really be an issue as its basically built in. That leaves VNC for which there are a ton of popular client softwares for and I am not well versed enough to provide advice on.

If your on a PC you can use Putty to SSH, you already would have an RDP client and like above I cant help with VNC.

I am thinking about what the best method would be allow me to connect to these various machines with as little interaction on the staffs part.

For a full desktop view: VNC, various vendors on multiple platforms.

For a faster-updating view of one GUI application at a time: Secure Shell. Like this:

 $ ssh -X username@systemname appname

(This example only works between two Linux systems.)

Secure Shell is already installed on most systems, including Windows (it needs to be activated). VNC is easily acquired and works cross-platform.

For the older Windows machines, you can install a vnc server (as far back as Windows 95, I seem to remember). I don't think early versions of Windows ran RDP servers, but I could be wrong. I'm pretty sure they'll run VNC.

Linux will also run a vnc server as well.

Not 100% certain about the Macs. Maybe some Mac guy can confirm.

VNC is similar to RDP for remote administration.

I use both currently with a Guacamole server which makes connecting really nice, but you can achieve the same effect with Remmina as it handles both protocols.