Different? It's wonderful! Flight following is the radar service we'd have in Europe if there were ten times more money allocated to ATC than there actually is.
I filed an FPL from Los Angeles to the Grand Canyon (about 4 hours flying time). I took off, called SoCal and was told "squawk 1066 - seems appropriate for you guys" and went all the way to GC, handed over from radar unit to radar unit, 1066 all the way. GC even told me "your flight plan is closed, sir" when we taxied in.
As slim_slag says, the airspace belongs to everyone, and you are treated as an equal. I've had radar vectors though busy airspace in among 737s, and they've been steered round me. Imagine that happening in the UK! You wouldn't be allowed within miles.
The NM and SM thing is a bit of a mess: I had to learn it all for the FAA IR. I can't remember all the details, but basically met vis and distances on the ground are in SM, distances in the air in NM. You'll find 22nm quoted as the range of certain VOR facilities - it used to be 25sm but VOR ranges are now in NM.
If you're on flight following, don't bother to give position reports unless asked. First time I flew across LA, I did the "N455 is at XXX VOR, 6500 feet, estimating YYY at (time)" just like CAP413 tells you. After about the third call, SoCal said "You don't need to tell us all that stuff, we know fine where you are."
If you're flying in California, you absolutely need the up to date sectional and terminal area charts and the A/FD (airport and facility directory). That will allow you to find out, for example, which of the dozens of SoCal frquencies applies for the sector you're in. You can even work them out for your route beforehand.
There are "recommended" routings, especially in IFR, which it's a good idea to follow: again, all in the A/FD.
Flight plans in the USA are totally different from here. You need to stare at one, with the instructions, and learn how it works. You also need to know the "slant code" for your aircraft - probably /A if you hire in California.
You file your flight plan by phoning 1-800-WX BRIEF (a free call) where you can talk to a real human being who will tell you all you need to know. You can ask for different types of briefing - if you don't know the types, tell him what you're doing and ask for advice on which form of briefing. You get weather, NOTAMS, advisories, and all sorts. He'll take down your flight plan for you.
You activate your flight plan, if the tower doesn't do it for you, by calling the local FSS (details on the charts and in the A/FD). You MUST ensure it's closed after you land - it's a good principle ALWAYS to phone 1-800-WXBRIEF after landing to make sure. I once got the call from them to ask if I was OK because mine hadn't been closed. DON'T FORGET!