When are you traveling into the city? SoHo offers lots of shopping but sadly, most of it is chain stores. A walk on The Highline( depending on the weather) and visits to art galleries( if they are open post Sandy)in West Chelsea. Meatpacking, south of West Chelsea, has some trendy high end shops.All have good restaurants too.
Union Square area also for shopping.

Go to the beginning of Bleecker St.-Abingdon Square- and just walk along Bleecker, past upscale stores. Keep going past 7th Ave, keep going toward 6th ave. past all the eating places and keep going past NYU bldgs etc etc. Interesting st. and neighborhood.

There are scores of designer/owner stores between East 9th and East 11th between 2nd Ave and Ave A.

There are many things for a 13 yr. old girl to do. Nearby to Times Square is Bryant Park where there is a beautiful skating rink, right behind the 42nd st. public library. Skating is free, just bring skates or rent for about $14. When you're done, walk to Grand Central 2 blocks east, to take the #6 train to Astor Place. Don't forget to ask where the "whispering Wall" is while in Grand Central, right in front of the oyster bar. You will invariably see people facing into the corners of a four-way arch, seemingly whispering to themselves. If you stand in opposite corners and whisper to the wall, your friend across the way will hear every word you say! Fun! Then, welcome to the East Village. All through the neighborhood you will find mosaics on the light posts by the "mosaic man". Check out his story on line. Very interesting guy. Look for his work all through the East Village. You can try to spin the cube in Astor Place, check out the new buildings there, and just below on third (cooper unions new building, pretty funky facade). You can then go across St. Mark's Place. You may find some colorful language on some t-shirts, but just browse, shop, look, stare...well, try not to stare, and head east until you reach Avenue A. Loads of fun shopping, great people watching, and great food nearby. Mahmoun's falafels are cheap, ($2.50), good, and delicious. B&H Dairy is on 2nd Ave. between 7th & 8th St. Best blintzes on the planet. Order one cherry and one cheese, with sour cream. Delicious! One block up on second Ave. is one of the last Polish/Ukraine ham stores called "Meat Market". Take home some country ham & some garlic sausage. Bring a double bag, because everyone on the train will smell what you're bringing home! Russo's Pasta store on E. 11th St. between 1st & 2nd Ave. has great homemade pasta, and the best fresh & smoked mozzarella in the neighborhood. Say hello to Fidel. He'll always steer you right! They are on the same block as Veniero's Bakery, (think Italian and New York Cheesecake, (yes, with a capital "C"), Buffalo Exchange clothing might be a fun place for her to browse, and at the end of the block, Liquiteria, the best juice joint in town. These smoothies will knock your socks off! If you are looking for japanese food, you are also in little Japan. Ipudo has got incredible Ramen on 4th Ave between 9th and 10th st. is a small Japanese bakery/cafe with inexpensive, not fancy sushi, and hot food. Great banana-chocolate croissants. Sunrise Mart, on the 2nd floor is a Japanese Supermarket owned by the same people. Fun to browse, and maybe take home some green tea, some wasabi snacks and some ramen noodles. If you're around after five, and want sushi for dinner, Sapporro East on 1st Ave. & E. 10th St. Not fancy, good food at good prices. Yafa cafe is on St. Mark's between 1st Ave & Ave. A. Really fun atmosphere that you won't find anywhere else. All of our out of town friends think this place is a real hoot. Good, healthy, tasty food. On the north side of the street, down the steps is the most incredible macaroon place called the Macaron Parlour! They are unique, and spectacular!Turn up Avenue A to 9th St. and have a cup of tea, hot chocolate or coffee of your choice with a piece of really delicious, non-traditional rugelach. Sit outside if the weather's good and watch the neighborhood go by. Turn down Ave. A, stop at Alphabets for some fun retro shopping, (any thirteen year old will love this joint), then next door to "Ray's" for a soft ice cream. He's been an East Village staple for thirty years or more, I think. Tompkins Square Park is across the street too. Turn right onto 7th Street, and you will find many good places to tempt you. Butter Lane Cupcakes. WOW! Then up the block to a little arepa place, followed by Luke's Lobster, for an amazing Lobster Roll. Go across the street to Porchetta, for the most amazing pork sandwich you will ever eat. Chickalicious is on East 10th St. between 2nd and 1st Ave. It is the smaller, more affordable version of Chickalicious Dessert Bar. Then there's always 16 handles. Probably a thirteen year old's dream, or frankly, just about any age person will have a blast here. Really great frozen yogurt that you taste first, (ask for tasting cups), serve yourself, then add your own toppings. East 9th Street between 2nd Ave & Avenue A is fun for shopping too. Lots of small stores with fun, inspired things to sell. Avenue C and E. 9th St. is "La Plaza" garden. It's a neighborhood garden with recycled materials used in unusual ways. Check out the "flowers" on top of the fence. 9th Street Espresso is across the street if any grown-ups need a most delicious caffeine fix. Pizza - Artichoke Pizza. 14th St. between 1st & 2nd Avenues. You won't need anything else to eat after a slice. Get their special Artichoke slice. Sounds weird to some, but oh, it is good! You can also stray a bit north to 1st Ave. & 19th St. for some of the best bagels in New York at Ess-A-Bagel. Bring a dozen home. They store well in the freezer. (Slice them first!) You can also stray south to Rivington Street to Economy Candy. Loads of fun for grown-ups and kids, alike. Much more real, and affordable than that uptown candy bar... I realize most of my "fun things" are food related, but it is such a good way to get a great flavor of a New York Neighborhood. You can come back time and time again to the East Village. From 4th Avenue to Avenue C, and from Houston Street to 14th Street. Don't forget to come back in the spring and summer to Tompkins Square Park. Lots of festivals, people watching, and summer fun. Hope you come to our neck of the woods! Enjoy your day! (If you do go to theWest Village/Chelsea area and the High Line Park, be sure to go to Chelsea Market, on 9th Ave., between 15th & 16th St. It is a truly fun place to eat and browse. No other place like it in the city. Stray a little south when here to Jacques Torres on Hudson & King St. While the chocolate is expensive, an enormous, fabulously delicious chocolate chip cookie is worth the $3.00 price tag, along with the intoxicating smell of chocolate. You can also see them making the chocolate through the windows inside and outside of the store. Take home a canister of Wicked Hot Chocolate Mix. Heavenly!

Sorry, I omitted the name of the Japanese cafe. Panya, (Stuyvesant Place, go up 9th street, between 2nd & 3rd Ave. and stay on the south side of the street, where it angles south) is a small Japanese bakery/cafe with inexpensive, not fancy sushi, and hot food. Great banana-chocolate croissants.

NYCFudee, we must be neighbors.

One of my photos is now hanging at the entrance of Pickle Guys on Essex off Grand. It is of the owner making horseradish at Passover.
_____

Add to that the Essex Market a strange blend of uber hip food kiosks and Latino food merchants.

Motorino may have the best pizza in the City.

Nai makes sensational tapas.

A number of spots make their own ice cream and gelati
-Il Laboratorio de Gelato (Houston and Ludlow)
-Sundaes and Cones (10th Street off Third)
-Van Leeuwen Artisan Ice Cream (East 7th Street)
-Big Gay Ice Cream http://biggayicecream.com/

And if venture into the West Village Cones on Bleecker and Grom (an Italian gelato chain)

Golemtoo, we just might be neighbors. Not a fan of Van Leeuwen's. Best Ice Cream or Ice Cream Truck hands down is Coolhauss. Worth going to their site and following them. They tweet their location once they've found parking each day. Often near Union Square. Ice cream sandwiches on home-made cookies, not to be missed!!!

This is a a great thread! With a 13 year old, I would suggest going into the W.Village, have lunch at John's Pizza or The Garage and walk around there. You might consider also going to the 9/11 Memorial pools. Forever 21 is right around the corner and all 13 year olds who are shoppers will enjoy that!

Here are some ideas gathered on another thread. These are specific to a 13yo girl, but includes some ideas she may enjoy - http://www.fodors.com/community/united-states/nyc-10-days-of-unique-shops-stores-snacks-share-your-favorites-please.cfm

These AREN'T specific...

Wow. Thanks for the replies and all of the great ideas. I love all of the food recommendations, and especially the unique stores like Alphabets and Buffalo Exchange. We are actually going to head into the city this Saturday. The weather looks like it will be great for some walking around.

You guys are definitely making my life much easier!

This summer we made a trip to New York with 2 teens and a 12 year old.

We did take up Golem's recommendation for Sundaes and Cones, and it was very good. I was just with my son that day and we were visiting techie shops. However I could see that this area was one I would love to come back to some day.

On another day, all five of us combined a trip to the World Trade Center area with a girls' stop at the Century 21 department store. My teen daughter liked it and would have liked to spend more time there. She could actually afford it on her budget too. The boys were a little bored souvenir shopping, so we then took the subway to Economy Candy. This was a great place, with lots of classic candies from past eras. I should mention though that the surrounding neighborhood wasn't a great neighborhood for browsing and wandering like the Village. We moved on to the village for pizza at Joe's Pizza.

Hope that helps.

Century 21 (not Forever 21, sorry), is a tjmaxx type store and is hit or miss. The hits are a lot of fun though!