Just listen for a couple of hours, you'll start to understand what is going on if you try.
You should probably also read this:
https://www.skybrary.aero/index.php/SectorisationTry following a plane departing LAX.
You make sure you know its callsign and you can see it on FR24 or planefinder or
https://tar1090.adsbexchange.com/You'll hear it being cleared for takeoff, the plane is switched from tower to departure frequency.
This frequency will likely not be announced as it is given to the plane beforehand.
Frequency changes after that will be given to the airplane, so you can hear the new frequency.
You follow that frequency (requires some searching on liveatc).
liveatc doesn't cover all frequencies everywhere, so this can be quite difficult.
As a starting exercise, maybe try the following:
A plane taxiing at JFK taxies out on ground frequency (121.9), is switched to JFK tower (123.9 or 119.1), is then cleared for takeoff.
After takeoff it's switched to departure (135.9 during the day, 125.7 when it's less busy).
The next frequency will depend on where the plane is going and the time of day.
But you should be able to hear that frequency and maybe find it on liveatc.
If you are tasked with researching this stuff, you should also understand altitude changes, heading changes and direct to.
https://skyvector.com/ can be helpful in looking for intersections.
Understanding departures, arrivals and approach procedures doesn't hurt either, read some plates here:
https://flightaware.com/resources/airport/KJFK/procedures