B737 FLT CREW INSTRUCTED TO FOLLOW ACR Y TO RWY 24C AT CLE STOPS AT WEST SIDE RWY 24L HOLD BAR AS DOES ACR Y; PASSING RWY 24C HOLD BAR.

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AFTER PUSHBACK FROM GATE; GND CTL INSTRUCTED US TO DO A 180 DEGS ONTO TXWY S (BEHIND US) SO ANOTHER CARRIER COULD TAXI BY FOR A FLOW TIME. SUBSEQUENT TAXI INSTRUCTIONS WERE TXWYS S; L; AND THEN FOLLOW THE OTHER CARRIER TO RWY 24C. APCHING TXWY U WE SWITCHED TO TWR AND COMPLETED THE BEFORE TKOF CHKLIST. THE OTHER CARRIER HELD SHORT OF THE RWY AT THE RWY 24L HOLD LINE; PAST THE RWY 24C HOLD LINE (BUT WE DIDN'T FIGURE THIS OUT UNTIL LATER). THE OTHER CARRIER WAS CLRED FOR TKOF AND WE CONTINUED TOWARD RWY 24C. SOMEWHERE PAST THE RWY 24C HOLD; BUT SHORT OF THE RWY 24L HOLD LINE; WE WERE CLRED FOR TKOF. AFTER TKOF MY FO FIGURED OUT WHAT WE HAD DONE. TWR MADE NO MENTION OF OURS OR THE OTHER CARRIER'S INCURSION. WE HAD TALKED ABOUT THE TAXI CHALLENGES AT CLE THE PREVIOUS NIGHT BEFORE LNDG. WE REVIEWED THE COMMERCIAL CHART PAGE. BOTH OF US HAD IT OUT DURING TAXI AND WE STILL GOOFED IT. THE INTXN OF RWY 10/28; TXWY Z; AND RWY 24L IS THE MOST CONFUSING OF ANY PLACE WE GO. WE GOT STUCKED IN BY 'FOLLOWING THE OTHER CARRIER' AND WATCHING THEM HOLD SHORT OF THE WRONG LINE. IF ANY PLACE COULD USE SOME WIGWAG OR IN GND STOP BARS; THIS IS IT.