After splurging on a new TV, one of the worst things you can do is to run it at the default settings. Because in my experience, they often aren't optimised to produce the best picture. Some of the better TVs may switch automatically to the appropriate settings, such as Game Mode, depending on the content. However, it's best to make sure everything's set up correctly — you only need to do this once. If you're used to watching TV at the default settings, these changes may take some adjusting. But trust me, these settings will help you get the most out of your TV. Note that the images below come from an LG TV — other brands will have similar, but not identical settings.
Turn off Eco mode

Using a power-saving mode helps with the environment, but at the expense of picture quality. Typically, a TV in Eco mode will be dimmer than desirable, especially for high dynamic range (HDR) content. And you didn't splurge on a good TV just to have it appear as dim as a cheaper model, right? But you may want to have Eco mode, for example, to reduce your electricity bill.
Some newer TVs also have a built-in light sensor to detect ambient light, and adjust the brightness accordingly, just like your smartphone. The fancier, more expensive ones have names like Dolby Vision IQ, which tries to maintain the impact of HDR content, relative to the ambient light. If your TV comes with such a feature, try it out and see if you like it.
Enable full bandwidth for HDMI ports

Out of the box, the HDMI ports on many TVs are often not configured to use their full bandwidth. Now this isn't an issue for standard definition content. But if you want to run games from a modern game console (PS5 or Xbox Series X|S) at its highest possible setting i.e 4K 120Hz, you'll need to turn on this setting. TV makers have their own names for it: Enhanced Format (Sony), HDMI Ultra HD Deep Color (LG), and HDMI UHD Color (Samsung). But I recommend going into the HDMI settings and enabling it for all the TV's HDMI ports.
Use Cinema or Filmmaker modes
Most TVs come with several preset picture modes with names like Standard, Vivid, Dynamic, Sports, and so on. But for watching movies, you'll want to change the presets to the ones called Cinema or Movie. These film-centric picture modes typically have a warm colour temperature, lower screen brightness, and are generally closer to how the filmmakers intend the movie to look.
Newer TVs may also have a Filmmaker Mode, which configures the TV to “preserve the director’s creative intent”. I recommend using this mode for most content if available. Here's what acclaimed directors say about Filmmaker Mode:
Disable sharpness and motion smoothing

While Filmmaker Mode quickly optimises your TV for movies, even this mode could be further tweaked on some TVs. For instance, my LG TV (in Filmmaker Mode) still keeps the Sharpness setting, which enhances the edges of objects in the video, at 10. Personally, I would change it to zero. I don't like that artificial look created by a high Sharpness setting.
Similarly, I would also recommend lowering or disabling motion smoothing, which is used to reduce motion blur, when watching films. Call me a purist, but traditionally, movies are shown at 24 frames per second. To me, adding frames (motion interpolation) makes movies look too much like real life. That's not how I want my movies to look. However, the realism or soap opera effect created by extra frames may suit some genres, like reality TV shows.
Enable Game Mode

If you're playing games on the big screen, you'll want the TV to cut down on all the extraneous picture processing, which adds input lag, and basically just work like a PC monitor. That's when you select Game Mode (if available) on the TV. For newer, higher-end TVs (usually those with HDMI 2.1 ports), Game Mode may also enable certain gaming features, like variable refresh rate. Some TVs also have settings such as Dynamic Tone Mapping or HGiG that affect the HDR in games. You should try these options, too.