I'm going to take a wild guess before explaining the correct way to connect everything. I suspect you're using the TV to switch between sources, rather than the receiver. The correct way, and please don't argue this, your sources (disc player, computer, streaming devices, etc.) should be...
That Denon should be fine in the price range you're looking for.
What speakers do you have? The TV really has very little to do with what receiver you choose, but the speakers do.
Also, what other components are you using in the system, and how are they connected? A lot of the time you don't...