The Embassy Suites is nice. Breakfast and a happy hour are included in the daily rate. It's walking distance to the track, bars, restaurants, and shops.
I would find a place away from the track. There are some Bnb in Saratoga Springs that might be good. Lots rent out homes during the horse season.
I agree with S Andreas. Keep an eye out on Airbnb, VRBO, HomeAway for homes for rent. Yes, there will be no daily maid service, however, you can't go wrong with the price, especially if you're outside of a mile away from track. Obviously more the closer you get. Now that there's Uber in Saratoga, you'll never wait for a ride.
I like to stay around Saratoga Springs . Numerous places to pick from, you can't go wrong.
Too many places to list. Depends on how much you want to spend. How close you want to be to the track. If your looking for quiet place to stay, you want somewhat away from the track. Union Ave gets pretty wild. Same with Caroline ave
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