Just had to take my 8 year old daughter to Mt. Sinai Hospital in Manhattan.
If you don't have some sort of EZ-Pass get one. The toll to cross the George Washinton Bridge is $12.50 with it, cash is $15.
www.e-zpassny.com
Try the SpotHero app for parking.
Try to move around on "off peak" hours. Tolls, etc. go up during rush hour times.
NYC parking meters take credit / debit / cash. There's one for each "section". Find the closest one to your vehicle, put the $ in, it gives you a little receipt that you display on your dashboard.
http://www.nyc.gov/html/dot/html/motorist/meterpark.shtml
And now to be 100% brutal and honest: Having grown up in Yonkers and the Bronx, you're a serious target with a van with Michigan plates, a 6 year old, and Grandma. Sorry to say it, but you're asking for trouble.
Leave the van someplace else safer and rent a tiny car with NY or NJ plates on it for your Manhattan time since you don't wanna bus / train, etc. Also, be ready to be trampled over walking on the sidewalk with Grandma and a 6 year old holding you back from the flow of people. Have your wallet in your front pockets, tell the girls no purses, walk around with your head held high like you own the place, don't look lost, and don't look like a tourist with your head craned up to see all the skyscrapers. When you're looking up someone's in your pocket, if you make eye contact with someone and look at the floor, that's a sign you're weak.
I hope this post is taken as helpful because it's meant to be. I'm just trying to have you mentally prepared.
It's not called the concrete jungle for no reason...