Well yes, there were amphitheaters all over the roman empire, even relatively small army cities would have their own amphitheater. It is also worth noting that the Colosseum, was by no means the first of its kind, quite the opposite in fact.

The first permanent stone amphitheater in Rome was the Amphitheater of Statilius Taurus, dedicated in 30 B.C., and by that time many other cities in Italy had already had permanent amphitheaters for a while. The one in Pompeii for example dates to 70 B.C. This was due to an old law that banned permanent theater structures within the confines of Rome. Towards the end of the republic people started poking holes in the law, to allow the construction of theaters. Mostly this was done by simply building theaters right outside the city limits, but this did mean that they pretty far from the center of the city.

Both emperors Caligula and Nero did build amphitheaters, but both of these were temporary wooden structures. And with the Taurus Amphitheater being damaged in a fire, by the time of the Flavians, the city of Rome desperately needed a permanent amphitheater, that could accommodate the large crowds of the city.

The Flavian Amphitheater was constructed in what had been gardens of Neros huge Palace. The site chosen was close to the Forum Romanum and right next to the colossal gilt bronze statue of Nero that had dominated the garden. The name Colosseum, actually derives from this statue, and has nothing to do with the structure itself.

Construction of the Colosseum seem to have been started by Emperor Vespasian around A.D. 72 and it was dedicated in A.D. 80, during the reign of his oldest son Titus. However construction seem to have continued well into the reign of his youngest son Domitian.

It is true though that the Colosseum with space to seat between 45.000 and 55.000 spectators was the largest of the Roman amphitheaters, but there were actually a few others that close to that size.

The second largest amphitheater, that of Capua. It is in a relatively poor state compared to the Colosseum, as you can see in this picture, but it used to almost as large. The earliest bits of the Capuan Amphitheater stem from the Julio-Claudian era, so it actually predates the Colosseum, but it was significant remodeled under Hadrian.

The third largest roman amphitheater, is a bit of an outsider. It is found in modern Tunisia, and is located at a site which is today called El-Djem, but used to be known as Thysdrus. Don't worry if you have not heard about this place before, the only thing interesting there is the amphitheater. As you can see in this picture, it is relatively well preserved and only slightly smaller than the one in Capua. It was construction around A.D. 238, and is thus significantly later than the other two. Unlike the Colosseum and the Capua Amphitheater, which were constructed with concrete and a stone facing, the El-Djem Amphitheater is almost entirely stone. This Amphitheater seems to have been largely a prestige project and unreasonably large, considering almost the entire population of ancient Thysdrus could be seated inside.

I hope that answers most of the questions you may have had.

Source: Roman Imperial Architecture, J.B. Ward Perkins, Second edition(1981)