Hello so I am CURIOUS as to how to get the best picture QUALITY out of my
new monitor, I was gifted a ASUS VN248 23.8" widescreen LED monitor
and had a few questions.

Whats the best connection to USE from my Graphics Card(R9 290) to my Monitor?
It comes with a DVI-HDMI cable, a VGA cable to DVI, and an MHL-HDMI cable

would I be better off with a straight HDMI-HDMI cable?
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How do I achieve the best picture quality out of my monitor?
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Not necessarily related to to picture quality, but when choosing wallpapers how do I find ones
that'll fit my desktop the best? (I don't know what aspect ratio)
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HDMI and DVI will be identical. HDMI-MHL adds some USB control features but that isn't utilized by graphics cards, and is more a hybrid USB/Display interface for devices like Set top boxes to be able to "control" the display.

Just don't use the VGA/D-Sub if you can help it. That will be an analog signal.

Your monitor is 1080p. that is 1920x1080. That is a 16:9 ratio. So for desktop wallpapers you want to find wallpapers that are ideally at that resolution or higher at that aspect ratio. You can use wallpapers at a different aspect ratio but they might stretch slightly if you use the "stretch" option. I use a 2880x1800 (16:10) wallpaper on my 2560x1440 (16:9) Monitor, for example, and it looks fine. (I use a Display PORT connection but it looks the same as with HDMI or DVI)