What's the viewing distance going to be? That will determine what size is appropriate. You'll definitely want to stick with 1080p on that spec rather than going above that though.
You also need to consider whether you care more about fast paced gaming and only rendered content, or whether you value image quality overall, including real life content as seen on the net.
An IPS or PLS panel will have better image quality, and they commonly have 5ms response, which is easily fast enough for most gaming, while TN panels are best for fast paced online shooters, but suffer on overall image quality and viewing angles.
There's also VA panels, which is what a lot of TVs use, but they can looked washed out in bright scenes and at wide viewing angles. They are making a comeback in both TVs and monitors due to having higher contrast ratio and brightness than IPS. This is largely due to HDR capable displays growing in popularity recently, which benefit from high contrast and brightness, but it only applies to 4k displays, which your 1060 isn't powerful enough for in most current games..
Regarding size, keep in mind the larger screen you use, the farther you will likely sit away from it. There are 32" monitors now that can be had in the $250 to $350 price range if you shop the sales, but many are not comfortable sitting within arms length from a screen that size. That means anytime you have to adjust the picture preset or color warmth, it becomes a hassle.
I sit about 5' away from my 32" display, and for that reason I'm glad it's a Panasonic IPS TV which I can easily and quickly make such adjustments on from the remote. Very few monitors have remotes, and usually the larger more expensive ones. Granted the main reason I need to make such adjustments is because games require more color warmth than TV broadcasts, but my game color settings usually don't look ideal on the net either. It's not as big a deal between net and games, but still enough to be annoying without the adjustments.
So while I may not have answered your main question, I think I have showcased most of the points you need to consider before making such a decision. If you prefer Nvidia GPUs over AMD, you may want to consider G-Sync, but it adds quite a bit to the cost. I would say you're probably going to want something in the 24"-27" range, 1080p, and something reliable. Dell makes some good displays. Try to Google for input lag specs on the model too. 10ms is very good, 20ms acceptable, 30ms tolerable for most, but slow for a monitor, and not good for fast paced online shooters.
Also, take note of the screen finish type, which is usually glossy or matte. The glossy ones have high transparency and offer best brightness, contrast, and image quality, but are very bad at reflecting tons of light. The matte finish ones are for avoiding light reflections but can look darker and not as sharp. So take note whether the room it's placed in will have any light sources in the back of the room, or whether you can possibly set up your lighting differently. I have two floor lamps with dark mica shades that cast the light up on the ceiling to either side of my display, and they have low watt LED bulbs that are 5000k, but not overly bright. 5000k-6500k lighting will make colors look more accurate because it mimics daylight better, but to a degree it can be too bright depending on the lamps you use and what the wattage is. This is why I go with 5000k vs 6000 or 6500, because you can get them in lower wattage.
I also have a 18" fluorescent tube lamp I place horizontally behind my TV in the middle of the screen area for games or movies that have a lot of dark scenes. When I do this I turn off the floor lamps and instead use a small table lamp in the back of the room with low watt compact fluorescent bulb and I place a piece of cardboard in front of it to block the light from reflecting off the screen. This yields just the right amount of ambient lighting both behind the screen and in the room for best contrast and detail in low light scenes, while having enough light to see the keyboard without reflections. It's pretty cheap to setup too. The lamp behind the TV is an under cabinet type with lens I bought at Lowes for $16. I put a cord on it by buying a 12' extension cord and cutting the socket end off. Then I just bought an inexpensive toggle switch I mounted near the plug of the cord to turn it on and off. All pretty easy to do, and cheap. I have the lamp sitting on a block of wood I had laying around.
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