In October 1938, in response to an urgent need to replenish the Soviet Union's ranks of trained intelligence officers after the bloodbath of Stalin's purges, the NKVD established a new training centre, the Special Purpose School (ShON). At secluded sites, new recruits to the GUGB's foreign intelligence department (INO; later to become the First Chief Directorate of the KGB) were trained for service overseas. This marked the first time that Soviet intelligence personnel were trained in groups, with a structured syllabus; previously, training was given individually and under strict compartmentalisation.

Recruits were usually university graduates (the Moscow State Institute of International Relations was a favourite recruiting ground) and Komsomol (Young Communist League) members, selected for intelligence and (perhaps more importantly) ideological conformity. Alongside the kind of training you'd expect — political and economic studies, intelligence tradecraft, languages — new INO officers were instructed in Western culture and etiquette by officers who had served abroad, to prepare them for what was almost invariably their first encounter with life outside the constraints of Soviet communism, and their first exposure to bourgeois capitalism. They also received stringent ideological tuition, to reinforce their commitment to Marxism-Leninism against the inevitable assault that life abroad would bring.

In 1943, ShON became the Intelligence School (RASH); later, it was reorganised as variously the (Dzerzhinsky) Higher Intelligence School, School No. 101, and the Andropov Institute. (Note — it's hard to keep track of these names, and their order, in part because every author seems to translate them from Russian slightly differently.)

Over the course of the Cold War, the KGB developed an increasingly sophisticated curriculum, lasting 2-3 years, for its intelligence officers. Here's an interesting remembrance of First Chief Directorate training in the early 1960s, written by Leonid Shebarshin, who went on to become the final chief of the FCD and (for two days in 1991) chairman of the KGB. Former KGB general Oleg Kalugin discusses his training in the 1950s in some detail in his memoir, Spymaster.

They're more general histories of Soviet intelligence, but Jonathan Haslam's Near and Distant Neighbours and Christopher Andrew's The Sword and the Shield include some interesting detail on the organisational evolution of KGB training over the years.

As for the historical accuracy of The Americans... that's another 10,000 character answer. As much as I enjoy the show, there's a lot of artistic licence in there (and that's fine!)

k1990 covered it pretty well, from what I can tell based on my own research into the subject. One interesting part of the training not mentioned was a peculiar game played as part of the training both recruits and counterintelligence officers.

As described by Oleg Gordievsky's Next Stop Execution: The new recruit was given a goal location, all of this occurred in Moscow, and before he/she set out they were to phone a number and tell people training for cointel the location they were starting from as well as the time. The cointel team would then have to practice following the recruit without being spotted. The recruit, on the other hand, had to perform a drop without being caught by the cointel team. The recruit failed more often than not.

This exercise had several purposes. The first and most obvious was to train cointel officers. The second is that it taught the recruits the best dead drop locations in Moscow, which made it easier to run agents. It also gave recruits practical experience in trying to lose tails, through a process known as "dry cleaning."

Moving on. It seems as if the KGB also stole recruits from the Soviet Military Intelligence, GRU, training school known as the Military Diplomatic Acadamy.

Sources:

Gordievsky, Oleg. Next Stop Execution: The Autobiography of Oleg Gordievsky. London: Macmillan, 1995.

Penkovsky, Oleg. Lt. Col. Oleg Penkovsky: Western Spy in Soviet GRU. CIA Freedom of Information Act Electronic Reading Room.