Sarasota: land of few practical stores
Luggage and bags: As someone who worked in the Sarasota / Bradenton airport for years, trust me when I say this: If your flying directly in, you do NOT want to check your bags.
Clothing/Shoes/Weather Gear: Even though it's freakishly sunny 80% of the time, it can be horrible when it does rain. Bring a slicker. Other than that, it's sandals, shorts, bathing suits, and sunglasses. Don't worry about looking too tacky or touristy. It's "the look" here to be spotted in a palm-tree ladden swim suit with pink and blue flip-flops on.
Toiletries and Medical Supplies: If you are trying to live a life free from poisons normally found in bug spray / sunscreen, I'd suggest bringing some from home. Organic / homeopathic products are very, very hard to find in Sarasota. If you do find them, they cost . . . a lot. Both are totally necessary.
Photo Equipment: Photo equipment, like everything else here, is crazy pricey. I'd stock up in Boise - or where ever you come from - where the prices are normal, if I were you.
Camping/Beach/Outdoor Gear: Camping stores are not too plentiful here, either. Myakka River Outpost (inside the State Park) can rent you most anything, but if you are flying down here to camp, I'd bring my own supplies.
Miscellaneous: Everything on the islands is totally overpriced. If you forget something, it's very worth it to trek back to the mainland and buy it there.