I used to recommend the Empire State Building, but it's now about $75 (I'm sure there are discounts somewhere).
Two things I've done in the past and enjoyed a lot more than I thought I would were the Circle Line tour and one of the open-top bus tours. Even though I know the city well, it gave me some new perspectives on things to see and do.
Walking tours are great, too. I particularly recommend The Bowery Boys (
https://www.boweryboyswalks.com/) as the guides are -very- knowledgeable and fun (The last one I did was Grand Central Terminal, and we had a great time). For the theatre district, there's Broadway Up Close (
https://www.broadwayupclose.com/inpersontours). The guy is a little too much, but he does know his stuff. There's also the Museum of Broadway, but I haven't done that one yet.
If you've got ancestors who came through Ellis Island, I recommend that (you also get the Statue of Liberty), particularly the Hard Hat tour, which takes you through the unrestored hospital buildings.
Two of my favorite restaurants are The Little Owl (I hate giving this one away), which is a tiny place in the Village with spectacular food, and is located on the ground floor of the building that was used as the establishing shot on "Friends." (When Matthew Perry died, they were inundated by flowers and people looking for souvenirs, in spite of the fact that they have nothing to do with the show.) The other is Emilio's Bellato on Houston, with the best chicken parm I've ever had. Pizza-wise, if you're in the Village, either Joe's of Bleecker Street (there's a branch on Broadway between 40th and 41st) or Lombardi's (there's usually a line and they sell only whole pies, no slices).
The Strand Bookstore is always worth a look, and will take you down to Union Square and the East Village.
Something I discovered when I was doing my shows there was that, staying in the East Village allowed me to really just walk around; like up Broadway (avoiding Times Square when I could) or up Park from 14th to the east 60s.
A lot of good restaurants in Harlem and on 9th in midtown. (I used to recommend just walking up and down Eighth, but it's gotten to be a little more touristy than I like, so I've gone west to 9th and 10th.) Just a ton of interesting-looking places.
The High Line is always great. I usually walk the whole thing, rather than break it into pieces. And the Whitney is at the south end if you're looking for another museum.
--Dave Sikula