Aussie,
I was never coached in doing it, so I've only done it one way, that's when I'm up in the air. Not sure if I do it to the book - perhaps they give me some slack with my pommie accent.
All you do is ask ATC for Flight Following. You don't need to file a flight plan or anything like that.
It's optional on ATC's part whether you get it - dependent on workload. But my experience on the busy eastern seaboard is that they will provide it more often than not.
If they do provide it, they will ask you your route, and give you a squawk code.
That's it.
When you get to the limit of their coverage, you get told to contact the next area, and your details are passed on behind the scenes.
On contacting the next ATC, all you say is something like:
"Newark Approach, Piper N-1234A with you"...
and that's it. Lovely.
One issue I have had, is sometimes the details don't passed on! You then get a puzzled Controller, asking you what you want. Suppose this is a workload thing.
Also, interestingly, even though you are flying VFR, on occasion you are given vectors and alitudes to follow. I guess this is the equivalent of a RAS in Blighty, but (correct me if I'm wrong), you need to have an IMCR at the least for RAS, whereas in the US you don't.
As this knowledge is home grown, happy to hear the definitive from anyone else.