What resort? It would be easier to answer your questions if we knew where you were staying.
For example, CSS at SSB has a grill and serves lunch everyday. They also have beef and chicken patties and do wonderful burgers, fries and jerk chicken. At CTI, since you are out on the Island, food is brought out to you at lunch time.
As for towels, the wonderful houskeeping staff leave 2 beach towels in your room that you can bring with you, and there are additional towels at SSB that you can use. You can discard wet or dirty towel and grab more. So there is no need bring your own beach towels from home.
It's an all-inclusive resort so everything is included (towesl and food too). The only things you'll need to bring are yourselves, your clothes and whatever things you think you'll need to enjoy a vacation in paradise!
Yes, you bring your beach towels from the room, and put the towels between you and the lounger / floatie / seat (except the seats at the swim up bar that are in the pool!) This is a general AN courtesy, to my understanding.
As for food, again, speaking from having been to CTI, they do bring out a small buffet lunch, when we were there it was two different meats, a veggie, and some desertish items.
You can bring towels and change them, if necessary, at the beach.
Apropos food, you don't mention which resort you are going to. CSS has a grill bar at the au naturelle beach (sunset beach) so you needn't leave the beach for a minute unless you want to. CTI has food brought over at lunchtime but I have no idea what sort of food as I have never been on the island. CN only has a bar and I think that it isn't manned at all times. I have never heard mention of a grill bar, or food being available there.