Venice certainly counted itself among republics; it was known as the "Most Serene Republic of Venice" to its contemporaries.
Venice is often studied in modern political science as an example of a city-state republic with an operating structure not unlike a modern constitution. [Here] (http://www.uni-mannheim.de/mateo/camenaref/cmh/cmh108.html) is a more comprehensive, but somewhat difficult to read (text-wise) history of the Republic. Importantly:
The Republic was originally a sea-Power whose merchants brought to her port the various products of Eastern countries, all de transmarinis partlbus orientalium divitias. The geographical position of Venice as the seaport nearest to the centre of Europe indicated her as a great emporium and mart for the distribution and exchange of goods; and, further, her situation in the shallow waters of the Lagoons gave her a monopoly of salt.
Cassiodorus, Theodoric's secretary, when describing the growing State, points to salt as the real riches of the young Republic ;" for men may live without gold," he says, " but no one ever heard of their being able to do without salt. Venice however required an outlet for her commodities; and this led at first to the establishment of factories in the districts of Belluno and Treviso, along the banks of the Piave and on one of the highroads into the heart of Europe (991), and subsequently at Ferrara (1100),, and again at Fano (1130).
And, in Venice, A Maritime Republic-- available [here] (https://books.google.ca/books?id=PQpU2JGJCMwC&pg=PA3&lpg=PA3&dq=Cassiodorus+venice&source=bl&ots=ZSuGI3wQ8V&sig=AMkIEixLWHme0wbdVxGWItV5NaY&hl=en&sa=X&ved=0ahUKEwjcgOjxg9bPAhXM6YMKHbg3B34Q6AEILjAC#v=onepage&q=Cassiodorus%20venice&f=false):
The life of the Venetians before 1000AD is and was relatively obscure, but a series of naval victories began in that year and came to a climax in 1204 with Venice's part in the conquest of Constantinople by Western crusaders... During the following centuries the Venetians as seamen maintained the wealth and the reputation of the Republic in the face of revolutions in nautical, military, and commercial technologies and trade routes. While neighbouring empires rose and fell, they elaborated a republican government in a form which aroused the envy of other Italian city-states.
Emphases mine, but it stands to reason that Venice governed itself as a Republic certainly from the 11th century AD and in a manner of pseudo-republic as a city-state prior to that. If you're feeling really nerdy, you can [read] (http://www.hpl.hp.com/techreports/2007/HPL-2007-28R1.html) a computer science analysis of how the Doge of Venice was elected-- an absurdly complicated governance process-- from 1268 until the fall of the Republic in 1797.
All of this to say: there were more than a few Republics between Rome and the USA. What might be another question worth asking is: why would you be taught that the USA is a historical exception of its era, comparable only to the heights of Rome? What message is that conveying to students about the significance of the USA in history?
The teaching of history in school is often a thorny issue because of the politics involved-- what is omitted is just as important as what is included.
As a follow-up question, Hamilton and Madison explicitly acknowledged the Netherlands and Venice as republics (Federalist Papers 20 and 39 respectively). Did later generations believe that these states were not republics? Did they simply overlook these two states (and similar ones like Genoa), or did the meaning of 'republic' evolve over time?
Just to make things really difficult, aside from the handy contributions from /u/whisperingmoon and /u/TRB1783, establishing that the US was not the "first Republic since Rome", I would suggest that Rome was never a Republic in the modern sense of the term.
There's a lot of debate over what sort of government 'Republican' Rome actually was. The term 'Republican Rome' is more a descriptive term of a defined historical period (after the fall of the monarchy c. 509 BC, until the permanent establishment of one-man rule under Augustus c. 27 BC, or perhaps as early as c. 60 BC or 44 BC if we consider Pompey or Caesar the first 'Emperor') than a real representation of Rome's form of government in this period.
The ancient Greeks (Polybius, building on Aristotle) considered Republican Rome to be a 'mixed' constitution, that is to be a somewhat equal mix of the three ideal forms of government: monarchy, aristocracy and democracy. However, this is a bit of a contrived attempt to force fit the round peg of Rome into the square hole of Greek political theory (Sommer 317-8, Seager 247-8).
Republican Rome is best characterised as a combination of aristocracy and oligarchy (Seager 247). In the modern day, especially in the context of the French and American Revolutions, a Republic is a form of government in which elected officials exercise executive power on behalf of the people. However, the Roman political system was organised such that the vast majority of people had no say in the election of magistrates, and therefore no say in how state power was exercised. Generally, only the richest members of society chose from an even smaller pool of rich aristocrats to determine who was elected to magistracies. Not only did the vast majority of landed citizens not get much say, much of the male population of the 'Republic' was not enfranchised anyway.
So I guess equating Roman Republicanism with American Republicanism is a bit misleading, aside from the fact that various forms of 'Republics' have existed in many places throughout history. The word 'Republic' has meant vastly different things throughout history.
Sources:
Sommer (307-18) and Seager (247-54) from Polybius and his world. Essays in memory of F.W. Walbank. Bruce Gibson and Thomas Harrison (eds). Oxford: Oxford University Press, 2013.
The US wasn't the first republic but it was probably the first large republic after Rome. The founders of the US were concerned about the longevity of the large republic they were setting-up and the last large republic was the Roman Republic. The Italian republics were city-states, the Netherlands and Switzerland were not that large and were relatively loose confederations, Poland-Lithuania still had a king, albeit an elected one. Maybe this was what was originally meant to be taught in your School.
A bit off topic, but I've seen medieval Novgorod described as a republic. How accurate is this?