I'm surprised no one has said "layers"!
You never can tell with April weather if it's gonna be cold, warm, or really warm. I always wear layers, year round.
I typically wear: jeans - either blue, dark blue, black, brown, or even red.
and, t-shirts: white, navy, black, red, etc. - solid, bold or darker colors; occasionally striped. Simple but elegant
Shoes: in the winter, black boots. In the warmer weather, black shoes or sneakers. In the summer, black slides/no socks
Over the t-shirt: a cardigan (I'm still bfing; gotta keep 'em accessible) and then a coat (and scarf, if need be). You can always stuff the sweater in your bag if you get warm, or hang your coat on the stroller (see below - storing stuff in your stroller).
In my front left pocket: my keys
In my front right pocket: money
In my rear right pocket: a "business card" wallet which I use to hold my credit cards, driver's license/ID, etc. - slim enought to fit in a lady's back pocket without too much bulk
If I have to carry a bag, it's either a messenger bag or a shoulder bag of some sort. I've used a backpack, too, but feel much more vulnerable to potential pick-pocketing (even though I NEVER worry about mugging/pickpocketing/etc. but BSTS, right?)
Stroller: sling if you can; lighter the stroller the better - and easy to fold up and get onto the bus or up/down subway stairs. HOWEVER - I have nothing against a nice slim (maclaren-style) double (side-by-side) stroller IF you are only walking. Nothing beats a place for both kiddos to sit and rest after a long day in Central Park. And with a one year old you may find yourself out-and-about during naptime and the nap can happen in the stroller. Also - where are you going to stow all your stuff/purchases as the day progresses? The stroller is your car/trunk. But, if you plan to hop a bus/subway/taxi, a single stroller with a sling or buggy-board is the way to go.
Things to do - I noticed you said you are staying near FAO - then you really must plan to spend at least two days in Central Park. There is a carousel, a zoo, a petting zoo, two museums, a skating rink, a marionette theater, rowboats, a little science center in a castle next to the Shakespeare Theater, an incredible Conservatory Garden up around 100th Street (across from the awesome Museum of the City of New York and the Museo del Barrio), not to mention the Harlem Meer, the Ramble, the Reservoir.......
Get a hot pretzel with that hotdog.......
Pack bottles of water.
Call me! I'd love to hang out in Central Park!
Oh, and I live near the other neighborhoods Meco was raving about - closest to Union Square, but that's also walking distance to Thompkins Square. Down here is the land 'o playgrounds and public parks. We also just got an awesome Whole Foods store.
Also check out these two Upper West Side Museums (great for kids):American Museum of Natural HistoryChildren's Museum of Manhattan
It's a great place to hang out!