If your TV model is a HDTV one, say a L37HDTVxxx, all you need is to get a TV antenna, then you can get the free air DTV signal. My antenna is a free one come with my previous TV purchase.
There is a good web site about air signal and antenna choose, I will post it here when I find the link, it basically tells you what type of antenna and direction you should have to get a specification channel, according to your home address.
According to pcmag review, ZIVIO 32" is doing very well for viewing DVD, even better than the comparing Sony LCD TV. Make sure you are using component video output, if your DVD player is not HD model.
I guess you are not use the LCD as a PC monitor, playing DVD from PC would produce even better result than playing from normal DVD player's component port, as it is optimized for 1368x768 resolution.
For LCD vs Plasma, you are correct, plasma is good at larger size. If you do want a model larger than 42", your don't have much choices except going plasma. Keep in mind that, the biggest drawback for plasma, it can not be used as a computer monitor, due to its burn-out effect. For this, I never give plasma a consideration, as I am a computer fan, I would even play games on LCD TV. Also, although larger size, the resolution for plasma is not higher, that makes you feel the dot is bulky. You'd better get a HD DVD player if you go with plasma.