(Edit to my parent comment: " It could be anywhere from one month, to six months" to certify on a single sector, but it's really more like one to three months.)
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All of the above.
Whenever an Air Traffic Controller transfers to a new facility, or even transfers to a new specialty within a facility, they have to train and get certified on all of the new sectors. Each sector is different due to traffic flow, types of traffic, equipment limitations, etc.
For the Houston Center Ocean specialty, equipment familiarization is very important for Controllers. There are 5 sectors: Ocean West, Ocean East, Offshore West, Offshore Central, and Offshore East. Ocean West and East deal primarily with aircraft flying between the United States and Mexico. There are different airways that aircraft can take, and each one has different characteristics (crossing airways, airways defined by RNAV fixes vs bearings off of VORTACs, radio coverage, and radar coverage).
In the Offshore sectors, radio coverage is harder to manage. There are multiple transmitters and receivers that are located on different offshore platforms, and the ocean elements and weather can affect the equipment. At our positions next to the radar scopes, there are touchscreens with many different buttons to select which frequencies we want to monitor, transmit on, use primary or backup sites, etc. Most sectors do not have to toggle between different transmitters, but in the Offshore sectors, that's a common occurrence. There are also different transmitter sites for the Ocean sectors, so we commonly get pilots saying that they're losing us on the radio when we are talking to another aircraft a hundred miles away, and we can hear the pilot just fine.
In the Ocean sectors, radar doesn't cover the middle of the Gulf of Mexico, so we have to rely on aircraft position reports unless they have ADS-B.
At HCF Center, there are mountains which block radar coverage, so it's good to know where we can expect to lose or establish radar contact with aircraft.
Yes, there is an element of becoming familiar with the routine traffic. You see many of the same flights every day, so you know where they are going. It got to the point at HCF Center, where if someone told me a flight number, I could tell them the departure and destination airports without looking.
However, just like pilots, Air Traffic Controllers cannot let routine turn into complacency. We can never just assume anything, if we are unsure, we have to ask or restate something. Safety is our number one priority.
The views expressed here are my own and not necessarily those of the FAA.