Hi, and welcome to info galore.
You can use the "find" tab to search for keywords like: cozumel (over 900 hits), mexican ruins (only 18), enter the ruins your interested in...chicken itza you can climb on...tulum you cant, but it has a beach where you can swim and see the ruins in the backround.
you can get to these places by tour, taxi, bus, or collectivo. Each choice is a little less expensive and a little more adventurous than the last. Tours are packaged and pricey, cabs are negotiatable but also expensive, a bus is cheaper but you have to get to the bus station that has the route to tulum, a little spanish is helpful to find it. lastly the collectivo is very cheap, they are minivans they run from playa del carmen to tulum all day, they stop and pick you up...you see them flashing thier lights if they have room and sometimes you have to cram in with your stuff on your lap, but they only charge like $5-$7 for the 70 km trip south to tulum...a bargain carpool.. but risky if you have limited time and have to be back before your ship sails.
taxis are at every pier and at the ruins..ALLWAYS negotiate the price BEFORE you get in the cab!!! Time is tough with a 40 minute ferry from cozumel to playa del carmen, then another hour and a half to tulum. You could spend about 5 hours just getting back and forth...i dont know how long it take to get to chicken itza but they are north of cancun which is 40 minutes north of playa del carmen...which requires the same ferry ride. So you might be better off with a beach, wait until you stop at a mainland port closer to the ruins of your choice to visit them.