What direction is one facing when one looks out over the fence? Is the patio in need of shade?

It looks like you might be CA, so I imagine you have to plant with wildfire protection as a consideration? With that in mind, are a fire pit and sparks an issue, or do you want it to be gas fired?

How large is the current paved patio? Could you combine the fire pit and the sitting area?

What room does the door lead into, and where is the kitchen from here? Is there easy access to the kitchen from somewhere in the patio area?

Would more paving, something like decomposed granite, or artificial turf fit into your plans? The current lawn looks fairly bad. Do you need a play area for kids or is this an adult only household that might not need a lawn?

Is there irrigation currently, and is that something you would want to/be allowed to continue?

Is there a HOA that controls how you can plant or design your yard?

If "BBQ area" means a place to park the grill, the only logical place right now is at the blank portion of the wall, which would mean rearranging of the furniture and possibly expansion of the patio. We can't tell from these pictures how it could be done. You need to supply a plan of existing features. And also reply to comments and questions. Shade is definitely in order. If a pergola was to provide shade, it would need cloth on it or a vine growing on it. Shade is definitely in order, as is expansion of the planting area. Have you checked to see if code will permit a firepit. It doesn't look like there's space to get it very far from the building.

If you are CA why don't you check out my profile for some ideas. we often put firepits off to the side so that it gives a destination to go to and bbqs are often close to kitchens so there is less walking. If you are in the Los Angeles area I design in that area and can give you ideas.

The below is one that we did that had a view and we created different spaces in the yard.

Studio City · More Info

To keep your fire pit low maintenance, go with a natural gas fire pit using something like fire glass or lava rock that won't require a lot of cleaning.