"New York vacation" is a tough concept for me to grasp. To me, vacation means getting as far away from the city (any city) as time and funds allow.
I can understand that.... but NYC in the summertime is a different animal. It empties out a bit and actually slows down compared to other times of the year.
Mudshark- As your NYC resident expert (and resident
), I'm happy to provide as much help as possible.
Regarding Yankee stadium- go as many times as you can, particularly if you've never been. The place is a relic in the same vein as Fenway and Wrigley. If the Mets were in town I'd say visit Shea as well b/c the place is also being torn down. You are absolutely fine taking the train up to Yankee Stadium- it's the best way to get the full experience. 3 different trains get you up there, so it's pretty idiot-proof. While the area is dicey as Rickster says, as long as you stay with the crowds you'll be more than fine. You'll never find a cab to take you home up there, and it'd probably be really hard to find one who'd actually take you up there b/c of the traffic.
As for lodging- if I'm thinking of the right hotel (is it the one on 6th Avenue) then it will suffice just fine. The nice thing about being on 6th is that while you're close to everything (including Central Park) but you're not right in the middle of tourist central. BE WARNED- all NYC hotel rooms are incredibly small compared to almost everywhere else in the modern world. If you can get a really good deal at one of the W hotels then I would look at those too, but there's very little difference btwn all of the chain hotels in NYC. If you want an amazing hotel experience then I recommend going balls-out and staying at one of the ridiculously expensive places- Sherry-Netherland, Mandarin, Pierre, Carlyle, Essex House, etc.
Food- everybody has made great suggestions so far. I would definitely take Strat's advice and go to Lombardi's- although be prepared for lines. I would not necessarily recommend going out to Brooklyn, even though Grimaldi's is ridiculously insane, because it's very easy to get lost coming out of the subway down there (not that it's unsafe, just really confusing) and the trip plus the wait could sully the experience. The Carnegie is great, although I personally prefer Katz's Deli,
http://www.katzdeli.com/, but in the end it's all the same stuff. I can make recommendations for pretty much every type of food you could ever want to eat in NYC, so let me know what you're looking for and how much you want to spend. If you want to have at least 1 insanely good dining experience (think minimum $300 for 2 people), I can recommend a ton of places that will redefine the way you think about food.
Other stuff- Since you will be there in the summer, make sure you hit up central park during the day. It won't be a typical weekend scene, but you'll get a great feel for how cool that place is. Most important thing is to do as much walking as possible.
If you end up staying at the Hilton, you can do one of my favorite walking tours of the city. Start at Broadway and 52nd st, the location of the Ed Sullivan theater, and walk south all the way down Broadway to the tip of Manhattan. Along the way you will pass/see/walk-through Times Square, the Chrysler Building, Herald Square, Macy's, the Empire State Building, Madison Square (be sure to get a break shake from the Shake Shack!), the Flatiron Building, Union Square, the New York Life Building, the West Village (take a quick detour to Washington Square Park), NYU, SoHo (lots of great shopping here), China Town, City Hall, nice views of the Manhattan and Brooklyn Bridges, Wall Street, Trinity Church, Ground Zero, Bowling Green and the Customs House, the Merrill Lynch "bull", views of the Statue of Liberty, Ellis Island, and a whole host of other places. End your trip with a detour over to the South Street Seaport to sit outside and have a nice drink as you watch the boats race up and down the East River. All told the walk is just over 5 miles, but it's the best way to really experience a large portion of NYC in an afternoon.