EXPECTING AN APCH TO RWY 25R AND CLRED FOR RWY 24R; APPARENTLY FLC ACFT WASN'T PROPERLY POSITIONED FOR APCH AND WERE DIRECTED TO GAR. THEN; AFTER ROUTINE APCH; THEY BECAME CONFUSED ON TAXI INSTRUCTIONS AND TAXIED ON AN ACTIVE RWY.

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SLEEP CYCLE: UP AT XA00; IN HOTEL BY XF30; UP AT XL00. FIRST TKOF AT XO00 FROM DFW TO OAK. TKOF FROM OAK AT XR30 TO LAX. LAND AT LAX APPROX XS45. WX AT OAK CAT II MINIMUMS OR WORSE. DUSK-FOG. MINDSET TO LAX WAS RELIEF/VFR/EASY. IT STARTED WITH THE HDOF TO LAX CTL. ATIS SAID RWY 24/25R SIDESTEP RWY 25L. WE AS A CREW PICKED/BRIEFED RWY 25R SIDESTEP L. MINDSET WAS FOR RWY 25R. APCH CTL DID NOT READ OUR MINDS; HOWEVER. BUT TYPICALLY THEY XMIT 'THIS WILL BE VECTORS TO ILS FINAL RWY 24R' OR SIMILAR SITUATIONAL RADIO CALL. IF THIS WAS XMITTED; WE DID NOT COMMUNICATE IT AND OUR UNDERSTANDING INTRA-COCKPIT OR TO THE CTLR. OUR MINDSET WAS STILL RWYS 25. I MADE THE COMMENT TO THE CAPT WHEN WE RECEIVED A 'HDG 060 DEGS OUT OF SMO VOR' RADIO CALL FROM APCH; THAT WE MUST BE GETTING VECTORS AROUND THE RWY 24 TFC. CTLR THEN TURNED US ON A BASE HEADING INTERCEPT FOR THE RWY WITH A R TURN. HE THEN CALLED OUT TFC TO FOLLOW. WE WERE ALSO ASKED IF WE HAD THE RWY IN SIGHT. YES: ON TFC; NO: ON RWY IN SIGHT. OUR LAST VECTOR PUT US ON AN EXCELLENT INTERCEPT WITH RWY 25R ILS. THE TFC TO FOLLOW WAS WELL AHEAD OF US AND ESSENTIALLY OUT OF SIGHT. THE CTLR THEN CLRED US FOR THE ILS RWY 24R. DUE TO RADIO TFC AND MY PUZZLEMENT; I ASKED FOR CONFIRMATION OF RWY 24R. IT WAS CORRECT AND WE CAME R TO 270 DEGS FOR A NEW INTERCEPT HDG. WE WERE CLRED TO 2200 FT MSL AND STARTED A SMALL DSCNT. I SET FREQS FOR RWY 24R ILS IN THE INSTS AND AS THINGS SETTLED DOWN IN THE COCKPIT; OUR SEQUENCING AND RWY ALIGNMENT WAS SUCH THAT THE CTLR COMMANDED A REENTRY AND RESEQUENCE TURN AND CLB TO 300 DEGS; 2500 FT. AN UNEVENTFUL ILS APCH AND LNDG WAS MADE. AS WE TAXIED CLR OF RWY 24R (L TURN) DOWN TXWY 65 WE WERE CLRED TO CROSS RWY 24L. FOR SOME REASON WE COULD NOT SEE THE TXWY 65 JOG TO THE R AND ITS LIGHTING ACROSS RWY 24L. WE TOOK A SHORT L ONTO RWY 24L THEN COMMUNICATED WITH THE TWR WHAT WE WERE DOING AND INTENDED TO EXIT ASAP ON TXWY 61. WE DID SO AND TAXIED TO PARKING. WE ALL LEARNED VALUE IN COMMUNICATING INTENTIONS AS WELL AS EXPECTATIONS.