Elliot - Caladesi and Honeymoon both have food concessions, bathrooms and showers. Caladesi has chair/umbrella rentals but not sure about Honeymoon. Keep in mind that Caladesi is accessible by boat only. There is a ferry that leaves from Honeymoon Island to go to Caledesi. You can bring chairs, coolers ...etc. with you on the ferry.
As for time of week, I would suggest avoiding the weekends if you want less people but it would depend on the time of year also. The Spring would bring a bit more crowds as well as some summer months. Our favorite time is October into November.
If you are into kayaking, try Sail Honeymoon on the causeway going to Honeymoon Island. If you are adventurous, you could actually paddle out to Caladesi instead of taking the ferry.
Here are some links for more info...
https://www.floridastateparks.org/park/Caladesi-Island
https://www.floridastateparks.org/park/Honeymoon-Island
http://caladesiferry.org/
https://www.sailhoneymoon.com/