Gravel or pallets for Bbq & picnic area be good or another setup?

Unless the pallets are pressure treated... they would only be a temporary construction...due to termites, dry rot etc. You could put them on top of a gravel base, use landscape/weed fabric first, and corral the gravel in a landscape border first.... and that would prolong their life some what..

DIYnetwork.com or HGTV.com or Pinterest.com will have lots of ideas on making a backyard patio... find one you like and go to it.

Gravel may not be good for under the cooker, tables and chairs due to the way they will move around and not be stable. The rest of the area would be ok for gravel. You would want something stable on top of the gravel for those areas for safety sake for yourself and your guests.

What a beautiful space! You are so lucky. I like the gravel idea as long as you create a border of some sort to keep it contained. I would draw sketches on paper of the features you want to have and think it through how you want to use the space and where to place things like a BBQ grill, smoker, prep table, beverage station, picnic table(s), etc. You might want to run lights out to the space for nightime use, maybe a fire pit.

Picnic area and flower beds!

If you do the gravel, get the pea gravel cause its a whole lot easier on the feet. I would also put some kind of border to keep it in place, like bricks or bender board. Be sure to put landscape fabric down first to prevent weeds coming through.

You could invest in a few pavers in the area where you want the BBQ so its on a level area. Picnic area I would just leave the grass.

You can use plastic pallets that have a solid top and make a deck area. I have my hammock on them. I found mine on craigslist. If you use wood, make sure you use sealant, because they will rot. With outdoor plywood, treated too, you have a nice deck and the good part is you can remove easy. If you r a dumpster diver, a lot of the tile stores throw out discontinued ,chipped etc behind their stores. I called and asked them if could get and they were happy to get it off their hands so they don't pay a disposal fee. You can put pots around it, leaving openings for access. Oops, put plastic down first to size. I use the black plastic rolls in the roofing aisle at Lowe's, it's actually cheaper than landscape fabric and nothing grows through it. This way you can configure your area any way you want. Get creative good luck.

Go with pavers for a level, stable surface.