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My TV (brought about a year ago) has a PC Input Socket - a VGA connection. So I can link my old laptop (which has VGA out) to the TV screen.
So as you have not brought the TV yet, how about buying one which has a lot of different options in terms of inputs? Mine has VGA, HDMI, SCART, S-Video, it has a USB port (not sure what that's for) and an Ethernet port - it streams iPlayer, though I do wonder what else it may be able to do with that.
Do you care that much about picture quality? The difference between some of these connection methods are to do with quality - HDMI for example is a digital connection, where as s-video and scart are analogue. The TV screen may be able to display 1080p resolution but your AVIs are probably not that high a resolution. They may only be 640x480, or 320x240. So expanding them up to a big screen may not make them look that good.
The usb is so you can plug in a memory stick or hard disk and watch content from there.
As for resolution, I watch stuff on my 40inch and it looks great. AVI files are not inherently of a lesser resolution. That is set by the person capturing the file in the first place. If one was so inclined, one can rack up the resolution to HD.