Assuming the best case scenario where this is a static shot and there’s nothing in front of the TV, you can get by with using the “Picture In Picture” plugin + color correction. So place the TV video above your main footage, add Picture in Picture, change mode to Free Form and move the corners to fit the TV. On top of that add Color Correction/Levels/etc. – whatever you prefer for color correction. This is because there’s likely some light falling on the TV, so it won’t be full bright, there might be a LUT on your main footage, etc. Basically it shouldn’t stand out too much or it will be distracting. Add some blur if it’s out of focus.
If the scene is not static, you could look into Motion Tracking, but that’s a bit more complex if you haven’t used it before. Though you could also manually keyframe the Picture In Picture corners if you really wanted to (actually wouldn’t take too long if it’s a linear movement).
If the TV isn’t a perfect rectangle you might have to add a mask either on your TV footage or your main footage. If you go with the latter I recommend you place your TV footage below main footage. Also I suggest you add at least a bit of feathering to your mask(s).
For best results you’d have to make this setup more complex than what I’ve described and in that case I might recommend you make this scene in something like After Effects instead. I reckon it’d be a better experience. But that depends on your setup, really. If it’s static you can get by with Vegas, but might have to use some track parenting and extra compositing modes for absolute best results.