I don't need to find a place to stay in Saratoga myself, so I can't give a lot of brilliant recommendations. Although the Gideon looks a bit on the stodgy side, it does have an excellent restaurant, and it's out at the Spa, so you can walk to the Performing Arts Center and back. Not that there are huge traffic jams for the ballet, but some might like the convenience.
Here's a link to a listing of some Saratoga-area places to stay:
http://www.discoversaratoga.org/asppages/L...sp?NAME=Lodging
I personally would probably love to stay at the Adelphi, as I've always enjoyed the cluttered Victorian look at the bar. Aside from that, I'd recommend looking for a place downtown, or near downtown, such as on Union Avenue, Circular Street (I would love staying at the Batcheller Mansion!), or, of course, Broadway.
Saratoga in July is rather pricey, but at least it's not August, and this year I don't think there's any overlap between the ballet and horseracing seasons.
There are lots of great places to eat in Saratoga. Personal favorites are Mrs. London's, very expensive, stunning coffee/tea/bakery place on Broadway, the Breadbasket (I think that's the name), a very inexpensive coffee/tea/bakery place on Spring Street, and Hattie's Chicken Shack, which would be worth the price of admission if only for the name.
There's a nice bar/restaurant at the Hall of Springs, which is right next-door to SPAC, practically, and gives one something to do during intermissions, or while there's a Diamond Project ballet going on. It is a shame they're not bringing Midsummer again, as SPAC is really the best place on earth to see this ballet. I'm so glad I caught last summer.
I don't know much about the Sheraton except it's at a nice location, is ugly as sin, and is probably quite comfortable.