I think the water fountains taste terrible, so we bring MIO water flavorings. I don't want to bring too much water in, but can't bring myself to spend $3.75 for every single drink all day. We start with a water bottle or collapsible "flask" nice and cold from the resort fridge (or, occasionally, partially frozen). We tried once to bring more than one per person, but that's just too much and doesn't stay cold long enough to be worth it anyway. I've finally gotten to where each of the kids is required to carry their own small bag if they want anything to carry (like autograph book, chapstick, pressed penny money) and next time we may consider insulated drawstring backpacks - I'm sure they won't be impervious to the FLA heat, but may keep our water bottles a little cooler while still not being very bulky.
Even though I knew about the free water at counter service places, it seemed we were always in a line when we realized we were thirsty or the CS lines seemed overly long and we bought something at a cart or refilled from a water fountain instead.