MD80 CREW TURNED OFF THE LNDG RWY OPPOSITE THE CLRNC AND THEN HAD A RWY INCURSION AT EWR.

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WE LANDED AT EWR ON RWY 4R AND TOOK THE HIGHSPD AT TXWY L. AFTER CLRING THE RWY; WE WERE GIVEN INSTRUCTIONS TO TAXI ONTO TXWY P AND TAKE TXWY J TO THE GATE -- OR SO I THOUGHT. INSTEAD OF TURNING L ON TXWY P AS I SHOULD HAVE; I TURNED R AND PROCEEDED TO TAXI LOOKING FOR TXWY J TO TAKE TO THE GATE. ASSUMING THAT WE HAD BEEN CLRED TO TAXI TO THE GATE; WE CROSSED RWY 29 THINKING THAT TXWY J WAS ON THE OTHER SIDE. AS IT TURNS OUT; WE HAD GONE THE WRONG WAY. WE STOPPED AT TXWY Z; SHORT OF RWY 22R AND CALLED FOR CLRNC TO TAXI ACROSS. WE WERE GIVEN CLRNC TO TAXI ACROSS AND TO THE GATE. NOTHING WAS SAID BY TWR OR GND CTL. WE DISCOVERED THAT RWY 29 WAS ACTIVE WHEN WE SAW AN ACFT TAKE OFF A FEW MINS ALTER. BIG LESSON LEARNED. CONFIRM THE TAXI INSTRUCTIONS WITH THE ARPT DIAGRAM AND STOP THE ACFT IF THERE IS ANY DOUBT. SUPPLEMENTAL INFO FROM ACN 547985: AFTER LNDG ON RWY 4R; WE TURNED OFF THE RWY. WE WERE TOLD AS WE TAXIED OFF TO TURN S ON TXWY P AND HOLD SHORT OF RWY 4L AT TXWY J. WHILE I DUG OUT MY ARPT DIAGRAM; THE CAPT TURNED N ON TXWY P. WE BOTH STARTED TO LOOK FOR TXWY J. AS IT DAWNED ON US THAT WE WERE HEADED THE WRONG WAY; GND CLRED US TO TXWY S; WHICH IS ACROSS RWY 9. THE TIMING WAS SUCH THAT WE WERE ENCROACHING OR VERY CLOSE TO ENCROACHING AN ACTIVE RWY WITHOUT CLRNC. GND DID NOT SAY ANYTHING AFTER THAT ABOUT THE INCIDENT AND CLRNC WAS RECEIVED BACK TO THE GATE. CONCENTRATION ON THE ARR AND APCH LEFT ME UNPREPARED TO DEAL WITH GND MANEUVERING.