About 20 years ago, a friend and I took what turned out to be a private bike tour of Central Park. It was a couple of hours, and the girl leading it showed us her favorite little places, like the Bow Bridge and the Ramble. Having a bike makes it possible to see a LOT more of the park than walking.
That said, I was poking around and noticed a "hidden secrets" walking tour. I know nothing about it other than reading the website.
https://www.centralpark.com/tours/walking/hidden-secrets-of-central-park/
If Central Park is a high priority for you, then using a bike to cover more distance and sights, and adding the "hidden secrets" tour or something like it, might be a good combination.
The same year I took the bike tour, I took a tour on one of the open-top double-decker buses, and when going down 5th Avenue along the park, the guide pointed out Harlem Meer and said it was the Reservoir. Argh. Suffice to say the park was best done with someone who loves it in particular.
As for burgers, in my opinion they are too variable to name a "best" because it depends on what a person likes, but an interesting place is Burger Joint in the Parker Meridian hotel on 56th Street. The entrance is in the lobby of this fancy hotel, behind a curtain, marked with a neon hamburger. Just burgers and fries in a very small space (best to avoid lunch hour), but interesting because it's just so odd that it's there.