If you plant a row of 3 wide growing things, or 5 slightly narrower things you can have a good screen fairly fast. Think about where you want the pool. If it will be off to one side, then add more plants between the pool and the neighbor's house.
If you budget larger plants (24" box or larger) plants then you will have something right away, though it might not be full coverage.
If you get too many of something then they will crowd each other, and try to grow taller to get the light they need, and highly likely go bare or at least thin lower down. You could do mixed species that tend to grow in different areas and will work when planted closer. One may fill in lower down while the other grows up higher before spreading out.
If you have more room then think about a double row of plants. Perhaps 3 -5 trees and between and in front of them some large shrubs, or vines on the fence. Again, the combination should provide faster coverage right away without the plants competing as much, so they ought to stay denser over the years.
A tree with the main branching starting around 5' high, and spreading with some upward angle would allow you to walk under it.
A vine on the fence may actually grow higher than the fence by at least a foot.
The future pool should be taken into consideration. As mentioned by DoGooder, you want plants with deep roots, not surface roots that you would have to cut when you put in the pool. That could kill the plants.
Here are a few ideas. All of them are evergreen, reasonably clean for a pool. I have avoided the fastest growing plants, because they are often short lived, or have aggressive roots that you do not want near a pool. Check compatibility for your zone:
Elaeocarpus decipiens- moderate rate, dense, easy to garden under. Grows as a spreading tree, though narrower for a few years.
Prunus caroliniana- Somewhat faster, dense, easy to garden under. Grows as a somewhat narrow shrub, eventually as a tree. You can probably find them in a large box well on their way to being a tree.
Citrus- many varieties. Moderate rate, very dense, a bit messy if you do not pick up the fruit, but otherwise neat. There are small and larger trees, and large shrubs (also dwarfs- but you want something larger) Oranges and Grapefruits are the larger trees, probably the best for the screening. Other Citrus stay lower and might fit in as 'somewhat smaller plants filling in between the trees'.
Magnolia- Many species and varieties of evergreens (not all species are evergreen), mostly trees. You will have to research which are available in your area. Most are not as dense, but fairly clean, easy to garden under. Leaves of most are so large they do not cause problems with a pool- easy to net out.
Viburnum- There are some species that grow as small trees or large shrubs. Some are a bit tender in my area, so I am not so familiar with them.
Pittosporum- Several species such as P. tennuifolium, P. eugenoides, P. tobira and others. Might be a bit messier than some, still, not too bad. Dense or airy, depending on species. Some have fragrant flowers. Make sure they are not hanging over the pool. These are more shrub-like or multi-trunked when grown as trees.
Feijoa sellowiana- Fairly dense, not too messy, though I would not put it right over the pool. Not too fast, though. May tend to be irregular in shape.
Vines: Almost all vines will grow a bit higher than the fence. Whatever grows in your area.
Most of the trumpet vines, Passion vines or Honeysuckle would be possibilities.
I would avoid Campsis radicans and Halls Honeysuckle- very invasive,
I am not so familiar with all the many tropical plants that will grow in Florida, but freeze here. If you take this list to a good nursery, and include the picture they may well have other ideas about good plants for your zone.
