I have heard a lot of bad things about Plasma, from my friends who know about tech in general and from my friends who have worked in electronic retail. I have actually heard some good things about DLP's, though I haven't used one myself, if you are able to find one for a good price.
Plasma, at least last time I heard, has problems with image burn and longevity in general. CRTV's still have the best picture quality, but LCD's are catching up. Look into DLP if the cost isn't prohibitive though. I can't tell you much about brands cause I don't follow the stuff. Samsung and Sony are two names that have always been good though. So my vote is with LCD to answer the question.
On the other question, it kind of depends on your interests. PS3 will actually have as good of FPS's as the 360 in a matter of months. Both consoles have pretty broad game bases, so I don't think you can go wrong either way. PS3 has the better holiday lineup, IMO, simply based on games. Some of MS's best games are also on PC, a weakness that I saw with the last Xbox as well. I wouldn't buy an Elite, I would buy a Premium if you were going for the 360. I don't think I need to tell you to stay away from the Core.
Reliability-wise, PS3 wins hands down. I can personally vouch for the system's solidarity. My roomate who has a 360 treats it like a princess because the hardware is very tempermental, i.e. games can scratch beyond repair if the system moves at all while it is running and it has major overheating problems. I have had no problems whatsoever with the PS3 and have not even heard of very many. It also has a safety turn-off feature if it gets too hot (don't know if 360 has this or not) which I unfortunately found out about when a family member closed the system up in an entertainment system without any air flow. It still runs perfectly though, so obviously the function works.
Online-wise, 360 is better, but for a price. It depends if you are willing to pay $60 a year. It is a good service, but the PS3's is catching up. It may not be free forever either, but I doubt it will cost as much as MS's. Anyways, it is free for now.
If you are looking to fully utilize your TV, the Blu-Ray option in the PS3 is a plus. You get the 5-free movie deal right now as well (6 if you buy it on Amazon). Once again, that is at your discretion. HD-DVD doesn't really stand a chance, so I don't think you have to worry about Blu-Ray losing the format war either.
PS3 has more pluses at the new price point, but the 360 is still a good machine. Check out the upcoming games if you are still unsure.
We had two bags of grass, seventy-five pellets of mescaline, five sheets of high-powered blotter acid, a salt shaker half full of cocaine, a whole galaxy of multi-colored uppers, downers, screamers, laughers…Also a quart of tequila, a quart of rum, a case of beer, a pint of raw ether and two dozen amyls. The only thing that really worried me was the ether. There is nothing in the world more helpless and irresponsible and depraved than a man in the depths of an ether binge. –Raoul Duke
It is hard to shed anything but crocodile tears over White House speechwriter Patrick Buchanan's tragic analysis of the Nixon debacle. "It's like Sisyphus," he said. "We rolled the rock all the way up the mountain...and it rolled right back down on us...." Neither Sisyphus nor the commander of the Light Brigade nor Pat Buchanan had the time or any real inclination to question what they were doing...a martyr, to the bitter end, to a "flawed" cause and a narrow, atavistic concept of conservative politics that has done more damage to itself and the country in less than six years than its liberal enemies could have done in two or three decades. -Hunter S. Thompson