I once upon a time tried to use a 720 TV as a computer monitor, and the picture was jagged and kind of dumb. If I get a 1080p TV does that solve the problem? What other things should I be considering if I want to use this kind of setup? It would be using an HDMI cable to connect.
The 720 should have worked just fine once you set your video card to the correct resolution for that screen (native resolution). Same deal with 1080. I currently have a 720 (hooked up with vga cable) with my main desktop and my giant 50" 1080 (hooked up with HDMI) as the secondary to watch movie on. They both look great.
I’m not an expert, but I believe dedicated computer monitors generally have a faster response rate than TVs. Putting the TV to game mode may fix that though. In terms of resolution 1080p should be fine, though that depends some on the size of the screen.
So the native setting is in the video card settings? Someone mentioned the “native setting” as being in the TV. This makes it query the TV and match the resolution and size?
I’ve had my computer hooked up to a 1080p TV since 2010. With the hdmi cable, they seemed to figure out whatever they needed to amongst themselves.
yeah sometimes the computers default drivers/programs will be fine. but id say look into either updating, or finding a new one to install. drivers kinda help the hardware communicate run smoother. i use nvidia. even though i know my graphics card needs a major upgrade soon, the driver helps it hang in there till i can afford a new one.
“Native resolution” means the resolution that the panel is designed for. For a 720p monitor, this would be 1280 x 720. If you set the monitor resolution in your OS to that, it should look better than if you set the monitor resolution to 640x480, 1024 x 768 or whatever.
So any tv should work and look good with an HDMI cable, if we find the right settings in the Display section of Windows?
Yes, just match the resolution to the resolution of the monitor. If it’s a 1920p monitor, it’s probably 1920x1080, or it could be 1920x1200. If you’re using Windows, it may say something like “recommended” at a paticular resolution. That’s the one to use.
It won’t necessarily “look good”, it will look as good as the TV resolution will allow. The thing is, you are used to relatively crappy resolution on your TV and really awesome resolution on your computer, so even at its best it won’t be as crisp and clear as your computer monitor.
The hard-core gamers have figured it out, with multiple screens:
The owner of the current computer - tv setup says there is a “native resolution” choice that comes up when the computer or the X-Box is connected. I found the manual online, and the word native does not appear in it.
Does anyone have an idea of what this would be? This is the setting the owner is concerned about in getting a replacement tv/monitor.
I remember using a TV as a computer monitor back in the 1980s, but I don’t think that’s relevant. 
What does the manual say about what resolution the monitor is capable of? If, as you said, it’s a 720p monitor, it should talk about 1280x720 resolution. Try that. (If you get a 1080p monitor, try a resolution of 1920x1080.)
“Native resolution” is the number of physical pixels on the screen. This will be advertised. 1920x1080 (also called “HD”), 3840x2160 (also known as "4K), etc. I’m not a gamer, but I’m sure the X-Box has many resolutions that it can support. An LCD can support resolutions lower than the native resolution (but not higher ones) but will look best at the resolution of the physical number of pixels.