If by "hometown" you mean "college town," well, there are several good places in Saratoga Springs to snag a good burger. There’s a Five Guys in nearby Wilton (get a friend with a car to drive you, or take the 472 bus) that has the most delicious burgers in the land. It’s all about the Little Cheeseburger with grilled onions and BBQ sauce. And the Little Cheeseburger is not little. It is quite large. A perfect burger to scarf before embarking on various weekend shenanigans.
If you want a FANCY burger, might I suggest Sperry’s on Caroline? They have this insane Kobe steak burger that comes with shoestring fries. Sperry’s is definitely a bougie place — what normal people call "side dishes" they call "accoutrements for the table" — so make your parents take you on Celebration Weekend, or jump in someone else’s parents’ car and make them take you. You gotta do what you gotta do.
Runner-up burger goes to anything at the Stadium Café. Solid burger material, but there are a lot of televisions in there and you don’t want to get distracted from…you know…eating.
Milkshakes? Hmm. Skip the milkshakes and get a bubble tea at Plum Dandy. They have many flavors. Chai is the best. Mmm, tapioca pearls…