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Since this is so long, I'll give you the TL;DR version first:

Restaurants were all over New York City and downtown Brooklyn by 1870. There were some 5000 restaurants in New York by that time, as well as countless street food vendors, and specialty shops, such as bakeries, ice cream parlors, soda fountains, beer halls, and more. In Manhattan, virtually anywhere below Central Park, you were no more than a few blocks away from some restaurant or ready-to-eat food of some sort. Further north, Manhattan became more suburban, and then rural. In Brooklyn, the restaurant scene was mostly confined to downtown Brooklyn, and the heart of the "Eastern District", i.e., Williamsburg and Greenpoint. The other boroughs--before the city was consolidated--were mostly rural, but the suburban Long Island City, in Queens, and Morrisiana, in the South Bronx, may have had a handful of restaurants. Among the ethnic cuisines available were English, German, French, and Italian cuisines, and to a lesser extent, Jewish and Chinese cuisines. American cuisine itself was the most widely available, with regional American cuisines available at different restaurants, and often on offer on the same menu--Southern cooking, and New England cooking in particular.

Now, the (very!) long version:

If you just wanted a quick bite, a lot of the same street vendors that exist in New York City today existed in 1870. There were fruit stands of all kinds, sometimes specializing in a particular fruit, like apples or oranges, but many fruit stand proprietors operated by stocking whatever was in season.

There were also peanut vendors all around the city, who also sold chestnuts and walnuts and other kinds of nuts whenever they could get them in stock. Candy vendors also operated throughout the city, as did soda water sellers--both by sidewalk vendors selling it in bottles, and at storefront soda fountains. The Aug 29, 1870, edition of the New York Times reported that the summer of 1870 was very hot, and the popularity of soda water made it "The Drink of the Period":

"To [the proprietors of soda fountains] the Summer of 1870 has been a blessed and prosperous season...and the extraordinary heat of the present season has taxed the inventive powers of soda-water manufacturers to their utmost...It is scarcely twenty years since the sale of soda-water was confined mainly to bottled soda, or mineral water, as it was sometimes called, which was put up in half-pint bottles, and confined to two brands--sarsaparilla and lemon. Pretty soon leading druggists began to supply themselves with urn-shaped fountains, from which they dispensed soda to their customers, with four or five different kinds of syrup, including sarsaparilla, lemon, strawberry, raspberry, and pineapple...[A]t the present time, it is not uncommon to find in the stores of our leading thoroughfares[,] marble fountains erected at an expense of $10,000 to $15,000, from which are supplied thirty or more different kinds of soda beverages, comprising every variety of fruit syrups, wines, essenses, coffee, and mineral and medicinal waters known to the confectioners and pharmacists...In order to retain their customers after the warm weather is over, the larger now furnish hot soda for the winter months, with coffee, chocolate and other syrups adapted to the season...

"...There are estimated to be about 280 soda fountains in drug-stores in New-York, and about 150 in Brooklyn. Besides these there are large numbers in confectionery stores, and a great many 'stands' in the street. The number of glasses drawn from each fountain varies greatly, some of the larger establishments, like Bolen & Byrne's, No. 169 Broadway, and Hudnut's in the Herald building [i.e., the headquarters of the New York Herald newspaper, at Herald Square, 34th and Broadway], draw as many as 5,000 glasses per day..."

Ice cream was another sweet treat sold on the street, as well as out of drug stores, and at specialty "ice-cream saloons". Bakeries dotted the city, and contemporaneous city directories list many bakers who no doubt worked at those bakeries. Bread, of course, was one of their primary items, but they also served sweet pastries--cakes, fruit pies, custards--along with others that had their roots in New York Dutch culture--doughnuts, crullers, and cookies. These were located in both Manhattan and Brooklyn; a "Help Wanted" ad in the Aug 16, 1871, edition of the New York Sun was looking for "An English boy to milk cows and clean pans at Jarvis's Banbury cake bakery, 257 Broadway, corner of 8th St., Williamsburgh [in Brooklyn]".

Some pastries would be sold at stands on the sidewalks as well. Bakeries, ice cream shops and stands, fruit stands, peanut stands, and the like were located all over Manhattan and downtown Brooklyn, just as they are today, with various newspaper articles drawing attention to the abundance of street vendors around City Hall Park, Central Park, and the area in the vicinity of the South Street Seaport.

According to a description in The Great Metropolis: A Mirror of New York written in 1869 by Junius Henri Browne, street vendors also populated the length of Broadway in significant numbers, "border[ing] the great thoroughfare from Morris street to Thirtieth, where the throng lessens into a line." Browne called attention to the vendors in the area of Park Row and the Bowery as being among the most affordable in the city, but also the worst:

"Park Row and the Bowery are favorite localities for street-venders of the cheapest sort. They offer every kind of low-priced article, from a dog-eared volume to a decayed peanut. They furnish impromptu dinners and breakfasts for a shilling; prepare oyster-stews while you take out your pocket-book, and bake waffles while you determine the time of day. They dispose of frozen custards and sour milk, sweetened, for ice-cream; soda-water without gas; lemonade without lemons; songs without sentiment; jokes without point; cigars innocent of tobacco, and all manner of shams..."

If you were looking for something more than just a snack, then the city had plenty of restaurants to choose from. This was a period, in fact, where the modern sense of the word restaurant was taking shape. Before 1850, a "restaurant" had referred to an eating establishment that served French food, specifically, French haute cuisine. But by 1870, all sorts of different eating establishments that had once called themselves "eating-houses", "coffee-houses", and "taverns" were now billing themselves as "restaurants", "cafes", "lunch rooms", "dining rooms", "saloons", and other fancier-sounding names. Browne's book claimed that in 1869:

"The City contains five or six thousand restaurants and eating-houses of different kinds...They vary as greatly in their appearance and prices as in the character of their patrons. They range from the elegance and costliness of Delmonico's and Taylor's to the subterranean sties where men are fed like swine, and dirt is served gratis in unhomeopathic doses...

"...To a stranger, New-York must seem to be perpetually engaged in eating. Go where you will between the hours of 8 in the morning and 6 in the evening, and you are reminded that man is a cooking animal. Tables are always spread; knives and forks are always rattling against dishes; the odors of the kitchen are always rising. Is the appetite of the Metropolis ever appeased? you think. Whence come all the people to devour all the food that is displayed in every shop between Whitehall slip and Central Park."

The May 12, 1866, edition of the New York Tribune claimed "somewhere near 500,000 walking appetites invade the saloons to breakfast, dine, sup, or lunch" primarily from Manhattan and Brooklyn.

(For context: The boroughs were not consolidated yet. New York City was only Manhattan. There were about 1.5 million people living in Manhattan, with another 400,000 living in Brooklyn, which only included the north side. The south side of Kings County, below Prospect Park, including Flatbush, Gravesend, and New Utrecht, were still separate towns, and still rural, with suburban downtowns. In Manhattan, the Upper East and West Sides were residential, and more or less suburban, while north of Central Park, the island was still mostly farmland or undeveloped, with suburban pockets here and there, particularly along Broadway. Harlem was just starting its transformation from rural to urban development. Long Island City and Flushing in Queens, and Morrisiana in the south Bronx, were legitimate suburbs, while the rest of Queens, the Bronx, and Staten Island, were rural countryside with modest town squares. Those three boroughs combined had a population just over 100,000 people.)

The reason for all this eating out, according to the Tribune, and more succinctly claimed by Browne, was because of "the great distance between the business and residence quarters, and the consequent necessity of the commercial classes dining or lunching down town." Browne additionally pointed out that once the workday was over, a good three-quarters of all restaurants below Canal Street closed for the day. In that respect, the Financial District hasn't changed its character all that much.

Nor has midtown, or anything above Houston, with Browne writing the restaurants in the numbered streets "thrive, particularly toward midnight, after the theatres and the concerts and the operas are over. The up-town restaurants furnish quite a contrast to those in the lower quarter of the City. They have no confusion, no bustle, no jostling, no door-slamming."

A lot of these restaurants served cheap, American cuisine. Browne mimicked the overlapping orders of customers given at the downtown lunchrooms with, "Hameggs-for-two-oyster-stew-coff-and-ap-pie-for-three-porkbeans-ale-cigars-for-four-beef-steak-onions-porter-cigarfor-five-mut-'n-chop-mince-pie-black-tea-for-one".