Upgrade graphics card or monitor first

Definitely GPU first. The only slight negative to that is if you buy too soon, you may miss best deals if the prices drop further. I'm not sure that's a realistic concern though because the rumors of the Nvidia 1000 series no longer being in production have found to be incorrect. I also just found an article with Nvidia's CEO Jen-Hsun Huang saying the new GPU line won't launch for "a long time from now". This combined with AMD's new GPUs probably not launching until 2019 likely indicates the new Nvidia line will debut in 2019 as well. Plus many are speculating they will be priced high.

https://www.pcgamesn.com/nvidia-computex-gtx-1180-release

I went with a EVGA 1080 SC when they dropped down to $600, which is $50 less than they debuted at. They are down to $550 right now at the same place I bought mine, and will be at that price through Jul 7.

https://www.bhphotovideo.com/c/search?Ntt=1080%20SC&N=0&InitialSearch=yes&sts=ma&Top+Nav-Search=

I do not regret at all buying mine for $600, because as far as I can tell, it's chip was from a high binned early batch run and is very efficient. It idles at 33-35c and never goes above 65c at load while gaming.

What I'm saying is, while you may or may not find better prices later this year, if you wait, you might also wind up with a lower binned chip that leaks current more and runs hotter. That is called low ASIC Quality.

Nvidia for some reason blocked 3rd party tools (such as GPU-Z) from reading ASIC Quality on the 1000 series, but such things matter. My last GPU, a Sapphire 7970 Dual X, had a very low ASIC Quality and was horrible at overclocking. In fact the last year or 2 I had it, it wouldn't even run at it's 2nd factory speed of 1000 MHz. It would artifact badly if I didn't use the 950 MHz setting. One of the two fans also burned out and I had to buy a new set from China on eBay for $12.

The EVGA 1080 SC has just a reference PCB, but that also means you probably won't risk compatibility issues. It has a very nice full length backplate and longer life ball bearing vs sleeve bearing fans. The fans also don't turn on until it hits 55c, so it's very quite while on desktop and the fans should last a very long time. It has just a single 6-pin power connector and stock PCB power circuitry, so probably not the best overclocked, but it's also priced low and comes with one of the highest factory clock speeds of any 1080.

On the monitor, I would look at some 120Hz ones in a store, preferably one hooked to a PC with a game demo or console, or bring in a console and ask them if you can hook it up. However there are things related to smoothness of gameplay that have nothing to do with frame rates. Even the quality of CPU you have can affect how smoothly a game plays. There's also a free tool called Nvidia Inspector that has driver level optimization profiles for pretty much any game ever made, and it works better than GeForce Experience. What I'm saying is you may find 60 Hz is plenty good enough.

Lastly, don't sweat not being able to take full advantage of the 1080 right away, because you can use Nvidia's Dynamic Super Resolution on a 1080p display to simulate 1440p, or even 4k. I've used it myself and it looks way better than 1080p.

http://nvidia.custhelp.com/app/answers/detail/a_id/3587/~/how-to-enable-dynamic-super-resolution-in-games.