Best HDTV for non-HD content?

[Falconer]

Not sure if the question makes much sense, but every time I see a high-end HDTV in a store showing Standard Def DVD or TV footage, it looks appallingly horrific. Same goes for all my friends who have newer HDTVs. Colors are smeared, images are blurred, motion looks 100% artificial, etc. Videos recorded on my phone have more clarity!

Is this a problem that is standard across the entire HDTV spectrum? I would love to upgrade my TV in the near future, but I still have DVDs and TV shows that I enjoy watching, and I don't want to need to keep a vomit bucket beside the couch, for lack of a better metaphor. I need someone to clue me in and point me in the right direction.