It depends on your family. Do you have young kids? It could be a homework center. Like to read? Do floor to ceiling bookshelves and put a hair with a little side table and an ottoman so you have a good place to read and enjoy the books. Family game night is your jam? Make this the game center with cabinets of games. Have very young kids, make it a kids corner with a kids table for painting, drawing, etc. In a now area? Make it into a mud section with a washable rug so everything has a place and your floors don't get dirty from everything being brought in. Have teenagers? Ask what they don't have room for in their rooms that they would like to do together. Mind love video games and doing each others nails. 

Just a thought.

Where were you planning on putting the TV?

Here’s my first thought. The TV mounted on the wall with a storage console below. Cozy seating in front of the fireplace. A storage armoir or low cabinet depending on the stairwell wall being open or closed.

Who doesn't want to sit in front of the fire? But since it's next to the kitchen keep it casual. I envision an upholstered bench with a high back for warmth and comfort. Then have a separate zone defined by an area rug with comfortable seating around a coffee table that as suggested would double as a game table and large enough for puzzles, snacking, etc.

I dont see a single closet or storage for the front door area. I think you might find yourself sorely lacking in storage. and places to put clutter. Unless your just not including them on here. I also think formal living rooms are wasted space especially if your family room is already going to have 2 areas. I think you will find you need some buildins on that end and a coat closet.