The subway is the best way to get to the stadium and you will see lots of other fans riding the train with you so you need not worry about being alone on the train. There really isn't anyplace to eat near the stadium, it's in a part of the Bronx that just doesn't offer up the type of restaurant atmosphere you'll find in Manhattan.
You don't say which zoo you're going to but I'll presume you'll be at the Bronx Zoo, if you're spending the day in the Bronx. There too you would over-pay for plain old snack food but the Bronx Zoo is in a different area than the Stadium and if you ask someone at the information area, you may find that there could be some good Italian restaurants not too far from the zoo - maybe a quick bus or cab ride from the zoo. There used to be a LOT of Italians living near Arthur Avenue and surrounding areas and so you might want to try that but otherwise,
well, it wouldn't really be a true baseball experience if you go to famed, hallowed (Yes, I'm a Yankee fan) Yankee Stadium and have a really overpriced hotdog, ice cream and all the rest of the stuff! You get shagged financially but you'll have had the "complete experience" of being a fan at Yankee stadium
I think you will have a lot of fun, so long as you aren't a Red Sox fan!!

Enjoy the Yankee game and Enjoy visiting my home city!
PS: If you want to see a less expensive but still very fun baseball game, consider going to see either the Brooklyn Cyclones (a Mets minor league club) or the Staten Island Yankees (minor club of you guessed it) for a great night, more intimate surroundings, nice family atmosphere and more affordable everything. I would recommend the Cyclones over the SI Yankees only because you can add to your fun by riding the real Cyclone ROLLER COASTER (the team's namesake) which is still rated as having one of the steepest drops of all rollercoasters - you can ride two times in a row for a special price. Then, there are arcade games, more rides, go carts (I so did my best race car manouvering and led the whole pack!!) and Nathans, the most famous place for a hotdog in ALL of NYC. My Dad grew up in Brooklyn (and I live here now!) and all he ever wants when he comes back to Brooklyn is to have some Nathans Hot Dogs; then he tells me all the stories he's told me before about how he came to Coney Island all the time as a boy - yeah, Dad, you mentioned it once or ten thousand times, lol

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