Usually at the beach, a polarized filter is helpful, to cut down on unwanted reflections. In clear water it aids in being able to see to the bottom was well. Since you are in bright sunlight, the reduced EV from this filter does not impact your shots much.

To reduce washed out sky, you can try a graduated ND filter.

If you want 'silky water shots' in mid-day sun, you will need at least a set of stackable neutral density filters. Stackable will allow you to choose the best ND strength given the conditions, and is much more flexible than simply a 10 ND filter alone. You may find you need to use this in combination with a polarizer in super bright conditions. IMHO "silky water" is best done at twilight/sunset, but possible in day with the right filters.

As for style, I would get a screw-on polarizer, as you will need a circular polarizer for your DSLR, and you need the ability to rotate it to align with the sun. In bright sunlight it is often useful to just leave it on the lens during your shoot.

However, for a Graduated ND, I prefer the square slide in filters, as they allow me great control in aligning the graduations with the horizon, allowing me to slide the filter up and down in the holder.

For stackable ND, I choose the square slide in filters as well, as the typical filter holder allows multiple filters held at one time, so I can mix and match to get the effect I need. It also allows the circular polarizer to be mounted on the lens without any issues.

I have had good luck with Cokin 'P' filters for the slide in filters.