Hi Kim,
Thanks so much for visiting the Disney Parks Moms Panel with your question. You are correct,
Disney's Saratoga Springs Resort & Spa is a large resort. Part of the charm is that its large size makes it more relaxing and calm. Where you would prefer to stay within the resort depends on your unique family.
The Springs is closest to the main area of the resort, including High Rock Spring Pool, as well as most of the dining, shopping, and recreation at the resort. If you do not have a car, The Springs area can be nice because it has quick and easy access to these amenities. Since you have small children, this might be a good choice. The tradeoff is that because so many of the amenities are in this area, it can be busy with guests walking around. My family once stayed in The Grandstand. We enjoyed this because it was a little quieter, yet still had a pool with a pool bar for beverages or a quick bite to eat in the area. The Paddock also has a pool, and this one has a waterslide. The Paddock pool also has a quick-service dining location adjacent. Congress Park offers some quiet locations with nice views of Downtown Disney.
Overall, given that you have two small children, and if you do not have a car, and you do not mind a little extra activity and noise, I would suggest a room in The Springs. If not, The Grandstand and The Paddock would be my next choices. Please do know that while Disney will do their best to honor all requests, they are not guaranteed.
I hope that these thoughts are helpful for you, Kim. Have a great time planning your vacation, and please do not hesitate to ask if you have any further questions.