I have an Nvidia Shield and a Vizio M-series 65" TV. The TV is big enough that I think some VR stuff could be fun. I don't think the TV natively supports any 3D stuff, so I doubt normal polarized 3D glasses would work, but is there any more fancy hardware I could get? If so, are there any apps I could use?
No. VR requires specific hardware. The few VR apps that run on general hardware require very high-end phones with integrated accelerometers and even then the phone has to fit in a box with special optics. No way to do any of that with a TV and set top box.
It sounds like you may be thinking of stereoscopic 3D gaming on your TV which isn't the same as VR. VR always involves a display strapped to your face. Technically possible for the ShieldTV but I dont think NVIDIA is working on one. And no one is doing stereoscopic 3D gaming from consoles anymore. It is a PC exclusive thing now.
Best options for VR right now are Samsung phone with Gear VR, Google Pixel with Dream VR, HTC Vive on PC, Oculus Rift on PC, and the PS4 VR is coming out soon too.
If you dont want to switch phones and want to try out a cheap option (assuming you have an Android phone) you can get the Google Cardboard VR. Its only $15, not very good but will give you a taste of VR. Make sure your phone will work with it first, though.