it's been awhile since i stayed at a ritz carlton and went to the pool/beach, but i don't remember having to pay a fee to use their beach towels, etc. (if there is one, you can always bring down the towels from the room, just remember to bring them back. if you make friends with your housekeepers, you can get extra towels no problem...read on.)
in hotels where there's a resort fee (anyone know offhand if the rc charges one?) that typically covers things like towels and chairs.
it might be worth your time, thriftyfam, to call the rc directly (or just call rc reservations, since it's toll-free) and inquire about resort fees, parking, etc., so you can plan ahead.
in terms of tipping (i may not be the best person to answer this, but someone else will surely correct me if i misstep

):
when the bellman carries your luggage to your room, it's $1-$2 per bag. same when he takes it down to the car at the end of your stay.
when you pick up your car from the valet--each time--it's at least $1.
you'll find you get GREAT service from the housekeepers if you leave a tip for them daily, rather than at the end of your stay (i learned this here on ft) and several dollars a day up to five (are you guys messy?) should do the trick. if you're looking for more towels, shampoo, soaps or whatever, the tip to the housekeepers can pay off well here. we've gotten more than our money's worth in bottled water in some hotels with this--granted, this was in asia.
i'm guessing you won't be ordering room service....if you do, check to see how much of a service charge is already included...you may not need to add a tip at all, or just a buck or two.
this forum isn't really about saving money while at a luxury hotel, but here are a few thrifty tips. (be sure to read some of the other forums as well, particularly budget travel, as there may be some ideas there that apply despite your staying in a luxury hotel.)
you don't say how long your stay is...but you could find an internet cafe close to the hotel if you need information or to check email while you're in sarasota.
you might consider taking breakfast bars and/or fruit to be able to skip a hotel meal or two.
since i gather you're trying to save pennies, here's something you might not know--if you hide the soap, shampoo, etc when you leave the room (like in your suitcase or something) they'll replace it with shiny new bottles each day. when i moved up to staying at luxury hotels, i thought this was the most amazing thing, and used to come home with bags of goodies...until i had done it enough that said small bottles of shampoo, conditioner, lotion and soap threatened to take over my house. now i only do it if i really like the product...that ritz carlton soap is a keeper.
have a wonderful trip.
hey, one more thing...when are you going? i just checked out
tripadvisor and saw there's some construction next door...not sure that would be an issue for you, but it would for me! be sure to read more than just the first page of reviews.