In the USA, ATC will ask your destination if you are in contact with them, otherwise they do not know, even if you filed a VFR flight plan. Under flight following, all climbs and descents are at your disgression unless you are in Class B or Class C airspace, but it is a courtesy to advise them when you are changing altitudes. If they are busy, they may not hand you off to tower or approach control, and do not expect them to. They only do it as a courtesy when they are not busy with IFR traffic. If you bust Class B, C, or D airspace without a clearance it will be your fault, not the fault of whoever is providing flight following. Under VFR flight following, the center controller will not obtain any class B, C, or D airspace clearances for you.
By the way, if you're flying in the USA, this is all contained in the Airman's Information Manual (AIM), and much of it is also in the FAA's "Pilots Handbook of Aeronautical Knowledge" . Get them and read them.
You might also get a copy of "The Pilot's Reference to ATC Procedures and Phraseology". Very helpful
Corporate pilot (ex-airline), flight instructor, 22 years in professional aviation
· 1 decade ago