There is no way we can make any reasonable suggestions without knowing where you are located. What grows well in one area of the country may be entirely unsuitable in another.

It may also be that trees are not the best solution for creating privacy. A photo of the area would help.

Ditto. I don't know anything about palm trees because I live in the midwest!

I know a lot about palm trees, and they are way better than lots of tiny leaves falling in your pool. Some palms are much easier to maintain than others. But where you live will make a huge difference as to what you can grow. Some palms are not good screens; others make very fine screens. Tastes may differ, of course.

A photo (or better yet, several, taken from one spot and panning across the yard) would help along with your location.

With that size lot, large, and the money you put into your pool, you should get a landscape architect. Good luck.

some ideas

Ashley M, as a new home owner in California, I suggest you purchase the Sunset New Western Garden Book. It does list Palm as a good tree near pools. From experience I would never want a Washingtonia near a pool or anywhere on my property. I think trees would be of most use to block the view of two story house. The other fence sections only need shrubs. A lot of people avoid anything with thorns or attractive to bees near a pool. No one in my family was allergic to or afraid of bees so we had a star jasmine along the fence behind the pool. It made a pretty dark green backdrop.

Please post the photo's as requested by Yardvaark. Do not use panoramic view. Just pivot your feet. Include one taken from the fence looking back at the house.

It would be helpful to your planning to draw a to scale plan on graph paper of the yard area to be landscaped. Much easier to plan on paper before you buy all the plants. Palms are expensive so I would use them only where needed.

How do you feel about gardening? Do you want a low care landscape?