It's hard to believe that a new TV has a picture so much worse than an old TV. Even the cheap TVs look quite good these days. The things I would check: (1) Make sure the TV is not in "showroom demo mode". Many of them default to this mode and will be that way when you take them out of the box. Super bright, high contrast, ridiculous color over-saturation. Meant to grab your eyeballs in a hectic showroom. But terrible for in-home viewing. (2) Check for motion interpolation. Different manufacturers call this different things. Like "Smooth motion", etc. What this feature does is "make up" an imaginary frame of the picture. Say the picture normally changes 60 times per second (this is typical). Each change is called a "frame". The TV artificially creates and inserts a totally made up frame between each pair of normal frames. So you end up with 120 changes per second, of which half of those are artificial. This can sometimes look good on a fast moving sports broadcast. But it can really destroy the picture of a normal show. The result is called the "Soap Opera Effect" and you will see it abbreviated SOE. You remember how the old soap opera shows always looked different? Something about the lighting or whatever that made them really look like they were filmed on a sound stage? (which they were, of course) "Motion interpolation", "motion smoothing" or whatever each manufacturer calls it creates that same artificial look. It's funny, some people cannot see it. Others - and I'm one of them - can spot it immediately and we just hate how it looks. It takes a really good picture and makes it look terribly artificial and hokey. Things get even worse when you have a cheap TV trying to do this motion interpolation, and failing at it. (3) Check for a setting called "sharpening". Lots of times the default setting for this will be maximum. You want it to be minimum. Again, this is an artificial thing, trying to trick your eye into thinking the picture has more zing that it really does. This feature tries to create higher contrast along edges. It does this by artificially darkening one side of the edge and artificially lightening the other side of the edge. If you get really close to your TV screen (like only inches away) you will see that edges look to have little white halos around them, often noticeable on small moving things (like a person walking in the distance).
At first glance, all these features might seem to improve the picture. But they actually degrade it. Especially on the cheaper sets that may not implement the features very well. But don't fret - the cheaper TVs probably still have a pretty decent LCD panel, and you don't need those stupid features anyway. So turn them off and enjoy a better picture. It's the same thing as when you're auditioning different stereo speakers. The louder one will always sound "better" initially. This is your brain pulling a trick on you. Once you set each speaker so that your ear hears the same volume, you often times find that the one you initially thought was better, because it was louder, is actually a real dog. All the stuff I mentioned about the TV picture above is just making it "louder". More gaudy. More in-your-face. You get it home and you rapidly tire of that excess, and wonder why the picture does not look as good as you remember it in the showroom. Everybody loves watching the Fourth of July fireworks. But that's only once a year. 24x7x365 fireworks would have people hiding in their basements to get away from the bling-bling.
Warning: When you start turning off your new TVs bling-bling, initially you will feel that you're making the picture muddier and ruining the thing. Give your eyes and brain a chance to clear the artificial euphoria from the excessive stimulation though. After you make a change, walk away for a while. Go have dinner, come back tomorrow. Reevaluate the change through eyes that have been allowed to recover. A drug addict will always think things are worse when they're in the middle of withdrawal.