This depends on how you were instructed to change frequencies.
There three main ways this can happen:
Services Terminated ("dropped")
Bugsmasher 12345, radar services terminated, squawk VFR and try Podunk Approach on 123.45 for further advisories.
This happens to folks on VFR flight following when two facilities can't coordinate a handoff because of workload or some other issue. You're not "in the system" anymore so when you call the facility they suggested you're making an initial call like it's the first time you're talking to anyone:
Podunk Approach, Bugsmasher 12345 10 miles west of Smallville VOR three-thousand five-hundred level, request VFR advisories
or one of the many local variations on that request that controllers prefer.
You're going to have to go through the whole process of giving them your information (type, altitude, destination) again, because they have no idea who you are: You're just another VFR blip on the scope.
New Frequency, New Controller ("Handoff")
Bugsmasher 12345 contact Podunk Center on 123.45
This is probably the most common situation when you're talking to ATC: The controller is sending you to another sector that knows you're coming. They have your flight strip and know who you are and what you want, so your check-in is
Podunk Center, Bugsmasher 12345 three-thousand five-hundred, climbing to five-thousand five-hundred.
The new controller needs to verify your Mode C (altitude) squawk which is why that information is included in the call. Level, Climbing or Descending is also important (so if the Mode C is different than what you reported the next time the radar paints you they know why and don't have to consider it a Mode C error).
The controller will then give you an altimeter setting (because they're required to), and your flight continues as it did under the previous controller.
New Frequency, Same Controller
Bugsmasher 12345, change to my frequency 120.5.
This happens for a bunch of reasons - radio coverage issues or a bad RCO antenna for example, or if a controller needs to split the frequency for congestion. When you switch to the new frequency you let the controller know by saying
Podunk Center, Bugsmasher 12345 on 120.5
so they know you've made the frequency change.
This controller has already verified your Mode C (altitude) squawk and issued you an altimeter setting, so they don't need to do that dance again.