GA NAVION CFI AND SPI POTS ERRONEOUSLY ENTERED CLASS D AIRSPACE AS A RESULT OF MISINTERP OF CLRNC.

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INCIDENT OCCURRED ON A VFR DAY TRAINING FLT FROM OGD HINCKLEY ARPT TO J77 (SPANISH FORK; UT). I WAS THE CFI IN THE R SEAT FOR A STUDENT PLT WHO WAS FLYING. WE WERE CLRED FOR TKOF ON RWY 16 AND TWR ASKED OUR DIRECTION OF FT AS WE ROLLED ONTO THE RWY. MY STUDENT RESPONDED WITH; 'WE'RE SBOUND TO SPANISH FORK.' THE TWR THE SAID; 'ROGER. S OVER I-15 IS APPROVED.' THEY ALSO INSTRUCTED US TO MAINTAIN AT OR BELOW 5600 FT IN OGDEN CLASS D AIRSPACE. AFTER DEPARTING OGD; WE FLEW S OVER AND SLIGHTLY W OF THE INTERSTATE. IT WAS MY UNDERSTANDING THAT OGDEN TWR HAD COORDINATED WITH HILL AFB TWR FOR US TO FLY THROUGH HILL'S CLASS D AIRSPACE; SINCE OGDEN TWR HAD TOLD US 'S OVER I-15 IS APPROVED.' AFTER DEP; WE REQUESTED AND RECEIVED A FREQ CHANGE FROM OGDEN TWR; AND CONTACTED SLC APCH. AFTER 1 OR 2 MINS DELAY; SLC FINALLY ASSIGNED A BEACON CODE. BY THIS TIME; WE WERE WELL INTO THE HILL AFB CLASS D AIRSPACE. SLC THEN TOLD US WE 'SHOULD EXERCISE CAUTION TO AVOID HILL'S CLASS D AIRSPACE;' AS THERE WAS A LEARJET READY TO DEPART AT HILL AFB AND IT COULD HAVE CAUSED A TFC CONFLICT. WE ADVISED SLC APCH THAT WE HAD BEEN CLRED S OVER THE FREEWAY BY OGDEN TWR. APCH APPARENTLY RECEIVED A CALL FROM HILL TWR REGARDING OUR PENETRATION OF THEIR AIRSPACE; AND ADVISED US TO CALL HILL TWR UPON LNDG. I EXPLAINED TO THE TWR SUPVR AT HILL AFB WHAT HAD OCCURRED. HE EXPLAINED THAT OGDEN TWR HAD NOT COORDINATED OUR FLT WITH HILL TWR; AND OUR PRESENCE IN THEIR AIRSPACE HAD DELAYED THE DEP OF THE LEARJET.