I want to fly the LAX Coastal VFR Transition Route northbound, and I want to do the whole thing with flight following. When I get near Van Nuys airport, I'll need to tell ATC (flight following) that I want to change my direction and head eastward to a different airport. How do I do this? And more broadly, how do I tell flight following radar services that I want to change my destination? For some routes it seems easier for me to have them follow me with radar services to a certain airport that I don't intend to land at, then ask them to change as I fly to a different airport.
Just tell them. Like you told us in the question. Seriously, plain old English works just fine on the radio.
Don't get all chatty, and be professional, but don't overthink things and feel you need to use a bunch of radiotelephony code words to get a simple point across.
The most important thing is to clearly communicate intentions. Don't tell them you are going one place and then "change" it when you actually planned on going there all along. That's phony and unnecessary, and could confuse things if there any sort of emergency where knowing your planned route of flight or destination would be useful.
Simply tell them you plan to fly VFR up the coast to VanNuys, then turn Eastward and proceed to destination XYZ. Or, more briefly: "destination XYZ, via Van Nuys". It's just that simple.
And consider filing a flight plan.
If you need to unexpectedly change your destination, the correct phraseology would be "Facility, Cessna 12345, amend destination XYZ." They may ask you why, so have a concise explanation ready; "comfort" works for anything biological, such as needing a potty break, food or a break to stretch your legs. "Fuel" or "weather" are also common reasons. They don't really care what you say unless it sounds like an emergency (or may soon become one).
However, you should never plan to use this. If your route is within about 30° of direct, just fly what you want; ATC expects VFR flights to wander somewhat. If it's less direct than that, in your initial request use "via" the airport or navaid nearest where you plan to turn. So, rather than tell them you're flying to ABC and then amending to XYZ when you get close, just say you're flying to XYZ via ABC; it's easier on both you and them.