A quick rule of thumb, if you're on a discrete beacon code, then don't change frequencies unless instructed (certainly request it if you like).
As Kyle pointed out, ATC will provide radar service between the primary airport in a Class C and the edge of the Outer Area, which is an uncharted (but very real) 20nm radius around the primary airport within Class C airspace.
That is why you received the radar service. It wasn't because of your destination. However, if you've stated a destination, it can often be construed by ATC as a flight following request (otherwise...why say the destination? I'd suggest only giving a direction of flight if you don't intend to keep the radar service beyond the Outer Area).
Now, if you absolutely want to be dropped ASAP, then the magic words are "NEGATIVE RADAR SERVICE."
This will generally result in you staying with the tower, with an altitude restriction that will keep you BELOW the outer shelf of the Class C (do NOT confuse this with the Outer Area, which is a 20nm radius, well beyond the published Class C shelves, extending well into Class E airspace). Once you hit the edge of the inner core, tower will then drop you and have you squawk VFR, freq change approved.
Hope this helps. And yes, it's poorly taught by most CFI's, near as I can tell, which means that unless private pilot students do a deep(ish) dive on Class C airspace/services, these nuances will be lost without some reinforcement during training. It all gets VERY muddy, and you WILL have mixed results if you state a destination when leaving a Class C without explicitly stating whether you want FF. My suggestion, then, and to be clear, this applies to IRL just as much as here, is to EITHER state a destination with intended cruise altitude and a request for FF, OR you state a direction of flight and specifically say negative FF. That way, there is absolutely no ambiguity.
Per the SBA ATIS on our system...
V F R DEPARTURES CONTACT CLEARANCE DELIVERY, ADVISE ON COURSE HEADING, ALTITUDE, AND IF FLIGHT FOLLOWING IS REQUESTED. READ BACK ALL RUNWAY ASSIGNMENTS AND HOLD SHORT INSTRUCTIONS.
Since a destination is traditional when receiving FF, that would take care of the on course heading part. However, if you're not requesting FF, as you can see, there's no stated requirement in the ATIS to specify your destination...it actually muddies the water quite as half of the pilot population tends to think that specifying the destination is a proxy for requesting FF, whereas the other half think it's just a proxy for the direction of flight.
Also, be prepared, some socal sectors are so small, and so close to other Class C's that the SOP for the tracon is to drop VFR aircraft by default at the edge of the Class C, prior to the outer area, unless they have actually requested FF. For that reason, if you're not receiving FF, and you're flying within Socal tracon, don't be shocked if they drop you sooner than the Outer Area boundary.