So when and how are you handed over between the two.
Here's a typical departure from JFK on a flight to the west coast.
Takeoff JFK Departure frequency 135.9
About 5000ft NY departure on 124.75
About 10-12K feet NY ARTCC on 134.6
Passing about FL240-FL280 NY ARTCC on 134.32
Leveled out, then the ARTCCs just change every X miles depending on what sector. Each ARTCC is divided in to smaller sectors within it, just like states inside of a country. For example, here's the names of all the different sectors of airspace in New York Center with the frequency next to it:
NEW YORK ARTCC (ZNY)
Barnstable 135.8 125.92
Big Flat 132.2 133.47
Colts Neck 118.97
Douglaston 133.05 134.47
Elk Mountain 134.45 132.17 128.5
Flint Hill 135.75 132.1 134.6
Joliet 133.67 132.5
Matawan 125.32 127.17
North Mountain 133.5 120.97 118.45 128.57
Phillipsburg 134.8 132.87
Ship Bottom 134.55 133.05
For approach into EWR for example, it goes like this:
Washington center 126.87
Down to FL270 132.52
Cleared down to 8000' via the DYLIN arrival on that frequency,
then at around 8000-11000' switch to Newark Approach on 128.55 (that's the TRACON)
Then switch to final approach at 6000' for vectors to the airport.
EWR tower 118.3 on 5nm final to 20nm.
Here's a link to a map of the USAs ARTCCs:
Hope that helps!