Are you planning to stay in this house for a while? Some people want to have a separate dinning room (we have one and it rarely gets used) If you were to take that wall down, that would change the kitchen..do you want a larger kitchen? You could use that room for something else like a craft room if needed..it depends what you need and want to have

What else could you use the dining room for? Does it have any windows? Maybe it would make a nice media room. I wouldn't take the wall out unless you're up for expanding the kitchen. Show us all 4 walls of the dining room so we can help you make it useful to your family.

From the limited pics thus far..... Yes, I'd remove the wall and open the space, as well as expand the kitchen into the nook area. A large 10' island with four stools in the center for kitchen seating/ morning breakfast. Keep the dining space - just open the wall

wait,. Where is the fami,y room in reference to the kitchen?

A floor plan would be helpful. It sounds like it’s a good idea to remove the wall. Of course a professional would have to see if the wall is bearing and if it is, how to support the load from above and carry it down.

The dinning room looks pretty bright in the photo. The breakfast area seems dark (the photo was not loaded the first time I saw this), but it's probably the lighting you have there

Without a fllor plan to scale shoing all the spaces you are asking about impossible to say I will say that tiny island in the kitchen is useless so if opening the wall makes the kitchen larger I say go for it . Yoy will of course have to redesign the kitchen change the flooring everywhere electrical maybe plumbing big job so think it through

Without a floor plan, and some clues as to the make up of your family, flip a coin. For all we know, you need a dinning room. However, I've found keeping the din in the dinning room can be difficult without proper soundproofing.

mad, that's funny. Finally, a definition of dinning room!

I agree with the above comment. I'd rather eat at the dining table than the kitchen table, so the latter is the area I'd expand the kitchen into.

Knock it down and then what?

You don't need to knock the wall down. Just re-think how you use the rooms. The dining area is much nicer and more comfortable than your kitchen eating area. Use that as your only dining space, and use the kitchen area for a couple of comfortable armchairs and a small round side table as a place to have coffee, read, or just another place to relax. Or turn it into an office/homework space. You have lots of easy options.