Formally, Janus was worshipped in the month of (you guessed it) Januarius, though the circumstances of his observance are much less well understood than other Roman deities. The month is presumed to have been added very early in Rome's history to what had been a 10-month calendar, possibly due to Etruscan influence. The 9th of January was an important day of the year for Janus, when the Rex Sacrorum sacrificed a ram in his honor in the Regia. Rams are elsewhere associated with Janus (e.g. the records of the Arval Brothers). Janus had no official priest, and is most closely associated with the Rex (as above). The Rex also offered a minor sacrifice to Janus at the beginning of each month. Other days throughout the year enjoyed some observance of Janus, and many official rites of other deities required an opening call to Janus in his role as Beginner (primus); Vesta, in turn, ended the proceedings (ultima).

The most famous shrine of Janus Geminus "the Twin" was in the Forum, supposedly dating to the regnal period of early Rome. There is not much left of it now, but from coins it is clearly a narrow, lofty rectangle with doors at both end. Famously, the doors were kept open in time of war but closed during time of peace. Equally famously, the doors were virtually never closed during the Republic. This place was moved several times to make way for new construction, and was eventually redesigned as a four-faced archway some distance away from its original spot and dedicated finally, after some delay, by the Emperor Nerva. By this time, the schstick of closing the doors had become worn out from overuse by the Julio-Claudian emperors.

There was another formal temple of Janus in the forum Holitorium, not at present securely identified, but near the later Theater of Marcellus. It was built by Gaius Duilius in 260 BCE after his defeat of the Punic forces at Mylae. This temple, like nearly all the rest in the city, was restored by Augustus. It became confusingly associated with Portunus, another power associated with passages, and in fact had been dedicated during the Portunalia in 260 (August 17, but the day was changed to August 18 in the Imperial period).

A third shrine was constructed in the forum, near the Curia, in the 2nd century CE. Dio says it was made of bronze, or clad in bronze. I know nothing else about it.

The history of Janus is complicated and very obscure. His original domain was the doorway (ianua means "door" in Latin), but he also became associated with new beginnings and, like Vesta, liminal or transitional spaces. His day on January 9th marked the beginning of the festival we call the Agonalia, which apparently had nothing to do with him. It is his role as Watcher of Beginnings that seems relevant in this case. Like Saturn, Janus is a very ancient Roman force but also very obscure and very muddled by the time our sources get good enough. Ovid does not seem to know much more than us about Janus.

Informally, Janus was everywhere. There were numerous shrines and arches dedicated to him throughout the city, and every house or apartment block acknowledged him in some way and included daily rites to him along with the dozens of other localized deities. Every little bit of junk and masonry, every muddle pond, every dusty lane in the Roman world had an associated numen, and it was no different for Janus. Janus Patulcius watched over opening doors, but Janus Clusivius watched them close. There was even a "god" (more like a Shinto spirit) for the hinge, called Cardea, which Ovid seems to confuse with the enigmatic goddess called Carna. She protected doorways also, specifically against vampire owls. Check it out.

Like virtually all of the very earliest Italic deities, I suspect Janus' ultimate origins are from rural Latium or Campania. Here we delve into the Compitalia, which, briefly, are moving feasts which were put on at various locations in late December and early January, after the Saturnalia. Their origins are in the compita, which are the places out in the countryside where roads or paths crossed. The crossroads was always a place of superstitious power, and in earliest times the farmers would put on these portable feasts at the compita, where they erected little monuments (turres). Altars were arrayed around the monument, pointing towards different farmers' holdings. Some argue these multi-facing little structures were the original "temples" of Janus. There is no explicit linguistic connection, but the association of crossroads, passage, liminal space, and liminal time (the end of harvest) makes sense.

See, for instance:

Dumézil (1966) Archaic Roman Religion. Johns Hopkins.

North (2000) Roman Religion. Oxford.

Scullard (1981) Festivals and Ceremonies of the Roman Republic. Thames and Hudson.