CL65 FLT CREW REPORTS CONFUSION OVER RWY ASSIGNMENT ARRIVING SLC; RESULTING IN CLOSE-IN RWY CHANGE AND LOC OVERSHOOT.

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UPON BEING HANDED OFF TO APCH WE WERE ASSIGNED A HDG TO JOIN RWY 34R INTO SLC. WE WERE AROUND 40-45 MI OUT AND JOINED THE LOC COURSE. SOON AFTERWARDS WE WERE GIVEN A HDG TO JOIN RWY 34L. I QUERIED MY FO TO SEE IF I HEARD THE CORRECT INSTRUCTIONS WHILE I TURNED TO THE ASSIGNED HDG. THE APCH FREQ WAS BUSY; I WAS IN THE PROCESS OF LOOKING UP THE FREQ FOR RWY 34L FOR IT HAD NOT BEEN BRIEFED; RWY 34R HAD. (THE CTLR MENTIONED THAT RWY 34L HAD BEEN ISSUED WHEN WE INITIALLY CHKED IN.) IT TOOK APPROX 45 SECONDS TO A MIN BEFORE WE COULD GET CONFIRMATION ABOUT OUR NEW RWY ASSIGNMENT. IT ALSO TOOK ME AROUND THE SAME AMOUNT OF TIME FOR US TO GET THE NEW FREQ AND SET UP OUR FMS. IN THE PROCESS WE FLEW THROUGH THE COURSE FOR RWY 34L. APCH CTL ASKED IF WE HAD THE TERRAIN TO THE W IN SIGHT. WE DID AND WERE ABOVE IT. HE GAVE US ANOTHER HDG TO JOIN RWY 34L. NO GPWS OR TCAS WARNINGS WERE ISSUED. WE JOINED RWY 34L; WERE HANDED OVER TO TWR CTL AND LANDED UNEVENTFULLY. NO PHONE NUMBER WAS ISSUED TO CALL APCH. UPON SHUTTING DOWN OUR ENGS THE FO AND MYSELF DEBRIEFED ON WHAT HAD OCCURRED. TO THIS DAY WE FEEL WE WERE ISSUED RWY 34R UPON CHKING IN WITH APCH. THE PROB STEMMED FROM THE TIME IT TOOK US TO GET THE FMS AND NEW RADIO FREQ SET UP.