I don't have a bigscreen HDTV yet, myself, so take that into account when I give you my opinions. However, I am a geek.
Direct view CRT (same technology TVs have had for years) has a wonderful picture which will be bright and sharp. Deep blacks will be black, and bright colors will be vivid. However, they are big, heavy, deep, big, and at 30"+ can have problems with convergence. Though I love my Trinitrons, I don't think I'll ever buy another over 20". Burn-in can be a risk if you watch the same channel (with a logo bug) or play the same game a lot on it. They have the widest viewing angle of any technology.
Projection CRT is not a technology I've ever liked. Presumably they've gotten better at dealing with the convergence issues they used to have (basically there were three projectors back there, one for each color, and over time they would gradually stop lining up perfectly), but burn-in is still a risk. They've also gotten better at making black "blacker," but it isn't as good as a direct view CRT. They are deep, but not as deep as CRTs.
Plasma is a lie. Do not buy the lie. They're heavy, use more power than even CRTs, get extremely hot, have very poor contrast which gets worse as they age, and burn-in EXTREMELY easily. They're also very expensive. I do not like them, Sam I Am. They don't have convergence issues, though, and they're viewable from a wider angle than anything but CRT.
LCD direct view, such as computer monitors, is probably the way of the future. The picture is drop-dead gorgeous, with great brightness and tremendous contrast. Though they share the same form factor as plasma, they're lighter, cooler, use less power, and have a better picture. They do have a risk of getting dead (no color at all) or stuck (always red, green, or blue) pixels over time, but in general this is a fabrication problem and that's always getting better. The risk of dead/stuck pixels also limits their size somewhat. They have a somewhat narrower view angle than Plasma. The only thing I don't like is the price, which is rapidly falling, and sometimes it looks too sharp if you're up close and watching standard-def programming from a digital source (DVD or satellite). The great contrast and sharpness means every compression artifact shows up.
LCD projection (DLP) is my favorite current technology. Unlike CRT projection, there is just one lamp which projects through a single LCD panel onto the screen, so there are no convergence issues. Like CRT projection, you lose some contrast, sharpness, and viewing angle. DLPs are the lightest out there, and though deeper than LCD or plasma are far shallower than either form of CRT. They're sharp, but not as sharp as LCDs, so standard-def programs actually look better on them (IMO).
One important thing to consider, regardless of technology, is that the CableCard technology (which allows you to get a digital signal without an external box) is changing soon. CableCard 1.0 doesn't allow for things like live program listing or pay-per-view, and CableCard 2.0 won't be retrofittable. If you will be using a cable system without a DVR, CableCard 2.0 will (supposedly) allow for the most seamless use of their services. If you'll be using a dish, a TiVo, or a cable box/DVR combo, CableCard is moot.
jaaaaaaaaaaaaaaaaaargon