RGR's got great suggestions.
I work in LES and would like to add a couple more. Jadis on Rivington near Eldrige is a very warm and friendly place. They have a limited menu but the food is really yummy, dinner only. I had my girlfriend's birthday party in the back area and we had a blast.
A new coffee shop just opened on on orchard call Roasting Plant. They grind and brew your cup of coffee fresh. It's kinda fun to see the beans fly through the tubes overhead. This is just down the street from Il Laboratorio del Gelato. Which by the way uses the beans from Roasting Plant for their espesso gelatos. Yummm just had mint and dark cocolate on the bright and sunny monday.
If you want great muffins, walk into the essex market on essex between rivington and delancy and find rainbow fish and muffin man. They sell fish on one side of the stand and great muffins on the otherside. $1.25-1.50 of happiness. If you get there a little after 11am, you might get one just coming out of the oven.
If you're feeling for Japanese home cooking, you need to visit SOY on Suffork between rivington and delancy. I love their Spicy Tuna & Avocado Bowl. Their curry is also very good.
Good luck and have a great time while you're in the city.
A great place to stop in for a drink and a REALLY nice bite is Prune. It is a tiny place so you can have a nice drink and an app. at the bar.
Love both Babbo and Mesa Grill, though they are totally different. Mesa you can get a res. without too much difficulty or sometimes just walk in, where as Babbo it is about a month.
If you're new to the boards, you can trust RGR's recommendations without fail. For someone like me who does more lurking than posting, I often live vicariously through RGR's recs.
Anyways, for my two cents...for a real splurge (and I think also we need to understand what you mean by splurge because in NYC it can have very different meanings), I would suggest Per Se if you can get a reservation. The atmosphere and service is incredible and the food is beautiful and indulgent and amazing.
Another option is Babbo, which I admittedly haven't been to but gets plenty of great feedback on the board. Reservations can be tough there too but eating at the bar is a great option if it's just you or you and another person.
For a unique place to get some small plates and drinks, head to the lounge in the Mandarin Oriental Hotel at the Time Warner Center. It's on the 35th floor (or thereabouts) and has a fantastic view. Food is great as are the drinks.
Also at the Time Warner Center there is a Whole Foods and you can put together a picnic and go across to Central Park to enjoy.
Yet more options are Sala, a tapas place near Union Square on 19th between 5th and 6th or 'ino in the West Village (address is escaping me right now) for panini and other snacks and great wine.
For pizza in Manhattan with an old NYC feel, try Arturo's on Houston St (pronounced How-ston, not like the city). John's is also a popular spot on Bleecker but I'm not as fond of it.
Enjoy your visit!
Those two restaurants are really in very different categories - my feeling is that the only thing they share is really the fact that they have name brand chefs that are associated with the restaurants. I went to Mesa Grill once and wasn't all that impressed. If those are your choices then I'd pick Babbo.
To throw one more restaurant into the mix - Balthazar is a great place. Terrific French food in a loud and fun setting.
Without knowing your exact food preferences (which would be helpful), no surprise to regulars on this board that my splurge pick would be Eleven Madison Park. Chef Daniel Humm's cuisine is exquisite, the wine list is excellent, service is very cordial and professional, and the space is gorgeous!
As for places to grab quick bites, considering that there are literally hundreds of possibilities, knowing exactly what you are looking for would be helpful. For instance, there are many places where you can get a great burger. My favorite is Shake Shack. In addition to the burgers, there are excellent fries and delicious soft ice cream. Plenty of tables where you can relax and enjoy your food al fresco in lovely Madison Square Park. Best to go off-hours in order to avoid humoungous lines.
When it comes to pizza, the best of the best is Di Fara, which is located in Brooklyn. If you're interested, check the Outer Boroughs board. I don't have a favorite in Manhattan, so I will leave it to other Hounds to make suggestions.
Since it is your first time in NYC, you might want to consider taking my in(famous) Lower East Side eating "tour." It will give you an opportunity to walk around an interesting, historic neighborhood while sampling foods that are emblematic of NYC. I'm appending the tour here:
LES Food Excursion
For the quintessential NYC deli experiences, no place beats Katz's, on the corner of Houston (pronounced "how-stun") & Ludlow Sts. You're there specifically for the pastrami sandwich. When you enter, you will be given a ticket. Instead of opting for table service, do what the "natives" do and get on line for counter service. When you reach the counter, put a $1 for each sandwich in the counterman's tip cup and order pastrami on rye. He'll give you a piece to taste. If you like it (the best pastrami is juicy and has some fat on it), tell him o.k., and he'll make your sandwich, give you some sour pickles, and punch your ticket. Then, continue along the counter for sides – the cole slaw is good -- and drinks. Find seats at a table in the center of the room. (Tables along the wall have menus on them and are reserved for waiter service.) When you’re done, take your ticket to the cashier in front. It's cash only. Note: For the purposes of this tour, unless you have a gargantuan appetite, it would be best to share one sandwich in order to leave room for more tastings along the way.
When you exit Katz’s, turn left and continue along the same side of Houston St. You will come to Russ & Daughters, famous for all sorts of smoked fish and many other goodies. It's not a restaurant, but they make sandwiches to go.
After leaving the Russes, continue west a couple of blocks until you reach Yonah Schimmel's. Get a tasty potato knish, and make sure to ask them to heat it up.
Now it’s time for the quintessential NY drink – the egg cream. So, reverse yourself and head east on Houston until you come to Avenue A. (Note: Avenue A becomes Essex St. on the south side of Houston.) Turn left on A and head north until you get to the block between 7th St. and St. Mark’s Place. Look for a hole-in-the-wall candy shop, closer to 7th, with an overhead sign jutting into the street that says, “Belgian Fries.” (The place’s official name is Ray’s, but there is no signage to that effect.) One of the women behind the counter will make you a delicious chocolate egg cream.
When you’re finished licking your lips, go back to Houston St. and make a left (east) one block to Norfolk St. Turn right and walk down Norfolk until it ends at Grand St. Two places to look for at the corner of Grand and Norfolk: Kossar's, for freshly baked bialys (another very NY food) and the Donut Plant (self-explanatory).
Next, walking west along Grand St., you will come to Orchard St. Turn right. At 87 Orchard, snack on a pickle from Gus's World Famous Pickles.
Then, continue to 97 Orchard, b/t Broome & Delancey, where you will find the Tenement Museum. The tour will show you what life was like for immigrants to NYC at the beginning of the 20th century. ( http://www.tenement.org
Once you have finished the tour, Il Laboratorio del Gelato, right next door at 95 Orchard, is a must for some of the best gelato anywhere.
If your sweet tooth is still not completely satisfied, the final stop on this tour should do it. Continue ahead (north) on Orchard, crossing Delancey, then one more block to Rivington St. Make a right and you will find Economy Candy at 145 Rivington.
Note: It’s best not to take this tour on a Saturday since some of the spots are closed because of religious observance. Also, Donut Plant is closed on Mondays.
As I said, If there are any other foods that you are particularly interested in sampling, let us know.