I also drink a ridiculous number of cups of tea a day. The best is the Andapur Tea house in Asia, as the offer some great blends. Also tea place in China has a few unique ones like a jasmine. I personally take my own bags of tea, Lady Grey, Yerba Mate, Irish Breakfast, and many others with me. I then just stop at any of the coffee shops and get a cup of hot water. The Grand Floridian Afternoon Tea is a fun experience, but not my any better tea.
Outside of the World there are two great Tea Shops. One is inside any mall it's called Teavana, they have 40+ loose teas. And only in downtown Orlando they have a cute "hip" tea shop called Dandelion Communitea. It is great and the people are awesome. They also have food, but I have never tried it.
I am glad to know I am not the only one that has an assortment of teas in my purse at all times.