This previous answer of mine talks about the non-alcoholic beverages common in the colonial period in the U.S. The most common would be: coffee, tea, water, and lemonade. Another drink, called "orgeat," which was a "cold beverage made of barley, almonds, or orange-flower water", was also occasionally drunk. Hot cocoa was also around, though probably only on special occasions since chocolate was a treat. Other fruit drinks other than lemonade could also be made, mostly fruit juices made of apples, pears, or grapes, though it would have been more common to turn them into alcoholic drinks of ciders or wine.

In the 19th century, other drinks became available. Soda-water was invented in the late 18th century, though it and flavored soda-pop didn't really become available until about the 1810s or so. It was common by the mid-1800s, and quite ubiquitous by the 1870s and after. For example, 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..."

As you can see, there were many fruit flavors, and there was sarsparilla as well. Other such carbonated beverages had begun to appear throughout the 19th century, some of them early. According to the Oxford English Dictionary, "root beer" had begun to appear in print by 1815, "ginger ale" by 1818, and "sarsaparilla by 1844. The November 1, 1850, Journal of American Temperance, for example, listed common "temperance drinks" as spruce beer, sarsparilla, ginger beer, and lemonade.

The 1870 cookbook Miss Beecher's Domestic Receipt-book has a chapter on "temperance drinks", which includes several of the aforementioned: sarsparilla, ginger beer, coffee, tea, and lemonade (as "portable lemonade"). There are others as well, including an iced currant juice (as "currant ice water"), a virgin champagne substitute (called "sham champagne"), and "strawberry vinegar". Though it seems doubtful any of these were very common.

According to The Oxford Companion to American Food and Drink, phosphates also became popular in the 19th century, starting around the 1870s. These were similar to a carbonated soda, except with a "flat sourness". They became popular as a more masculine alternative to the sodas popular at the time that might add milk to concoction. The most popular flavors of phosphates were lemon, orange, egg (with a literal raw egg stirred into it), cherry, and chocolate.

Some of the bottled sodas that became available for purchase in the mid- to late-19th century are still around. Vernor's Ginger Soda dates to 1866 (or maybe 1880), Hire's Root Beer dates to 1876, and even Coca-Cola was first sold in 1886.

But you asked specifically about what a person would be able to purchase at a bar/tavern/saloon. This is a slightly different question, and a little bit more difficult to answer. If you include an "inn" in the answer, which often coincided with taverns, then a traveler no doubt would have tea, coffee, and water available to them. Possibly something like lemonade or apple juice might also be available occasionally, but that would be much less likely, and probably dependent on what the family who owned the inn/tavern was in the habit of making for their family, since they probably weren't requested too often.

But at a saloon that was in a larger city, there probably was little available aside from water being served at a saloon or bar. If you wanted coffee or tea, you'd go to the coffeehouse across the street instead of the bar. Bars would have a variety of liquers and cordials used to make mixed drinks, but they'd probably look at you funny if you wanted one mixed with plain water. Later in the century, soda-water would have been available as a mixer, but again, if you wanted a soda-pop, you'd go to the soda-fountain down the block, and not to the bar.

Which brings me to my other point. While the temperance drinks of lemonade, ginger ale, root beer, sarsparilla, and other soda-pops certainly were available in some bars/saloons by mid-century as a bottled drink, (1, 2, 3) it's unlikely they had soda-fountains in the bar. You'd need to go to the actual soda-fountain proprietor down the street for that, often doubling as a drug store, convenience store, or ice cream parlor. But moreover, some communities actually deliberately kept alcohol-selling bars/taverns from selling temperance drinks,(1, 2, 3) and had separate licenses for each type of establishment. This 1850 law in Quebec is one such example:

"...[N]o person who shall not be licensed to keep a temperance hotel or as an apothecary, shall vend or retail ay description of liquor known as a temperance drink, such as spruce beer, sarsparilla, raspberry vinegar, ginger beer, essence or juice of lemons or of oranges or lemonade, under a penalty of Ten Pounds for every contravention of the provisions of this Section."

This was likely done for a few reasons. First, perhaps to protect the economic viability of each kind of business. But second and more importantly, to keep the temptation of alcohol away from teetotallers, at a time the temperance movement was taking off. And third and related, probably to prevent bars/saloons from serving sugary mixed drinks that might be a bad influence, for the same reason "candy cigarettes" are no longer marketed to kids. It also worked the other way around: there were drug stores putting liquor in soda-fountain drinks, and licensing temperance sellers, and alcohol sellers separately helped crack down on the public nuisance.

This would have become more common in the second half of the 1800s into the early 1900s, as the temperance movement eventually achieved its goal in the U.S. with Prohibition. As one example of the "segregated" way that beverages were being sold, here is an 1884 newspaper ad for a billiards hall in Nebraska priding itself on selling "ginger beer" and "lemonade".

In any case, all those drinks were around, and it would depend on when in the 19th century, and where in the 19th century, you're actually talking about to say for sure. Ultimately, few people would really go to a saloon for a non-alcoholic drink, when there was likely to be a coffeehouse, or a soda-fountain down the street instead.

TL;DR: In the early 1800s, coffee, tea, water, milk, lemonade, and possibly hot cocoa. Fruit juices like apple juice and grape juice might also be available. By the mid-1800s, all sorts of varieties of soda-pops were available, as well as root beer, ginger ale/ginger soda, sarsparilla, and spruce beer/birch beer. Around the 1870s, phosphates were also available. Saloons did sell soda-pops, as well as coffee and tea. But in some communities starting around mid-century, alcoholic and non-alcoholic beverage retail was kept separate. If you went to a bar/saloon, you were likely to only be served alcohol. If you wanted a non-alcoholic beverage, you'd likely have to go to a soda-fountain druggist or to a coffee shop instead.