Mayan Ruins:

The three best are the easiest to see from Puerto Morelos. You could do Coba and Tulum in one day. Chichen Itza should be a day by itself or overnight.

Coba

Coba was a huge city begun in 600 AD with the tallest pyramid on the Yucatan. We rented a car for the day to get there. The ruins were very spread out so they provide bicycles to get from one ruin to the next. We also found it helpful to hire a tour guide at the entrance.  http://www.cancunsouth.com/cit_coba.html

 Chichen Itza

Chichen Itza is probably the most restored archeological site on the Yucatan. I know that you can take the bus from Cancun, but we rented a car and made it a two day trip. We stopped in Valladolid, an inland colonial city, Cenote Dzitnup, and Cenote Zaci on the way. We went to Chichen Itza for the night show and then stayed at the Hotel Delores Alba in Piste. It was a highlight for our teenage boys. The hotel has a pool that has limestone formations and tunnels that you can swim through. We went back to Chichen Itza first thing the next morning to see all of the sights. On the way home we made a quick stop at Tulum. Chichen Itza: http://www.cancunsouth.com/cit_chichen.html

http://www.internet-at-work.com/hos_mcgrane/chichen/chichen_index.html

This link is from a site that is a bit strange, but an extensive collection of good stuff on the Maya culture and history:

http://www.crystalinks.com/maya.html

Tulum

These are the beautiful ruins right on the beach that you often see in photographs. It’s a walled ruin with 60 well preserved structures from 1200 ad. You can drive or take a bus. It’s less than an hour south of Puerto Morelos. You can catch the bus at the highway. Tulum: http://www.cancunsouth.com/cit_tulum_ruin.html

Tankah Cenote

There are hundreds of Cenotes on the Yucatan penisula. We especially enjoyed this one. The turn-off from the highway is between Xhelha and Tulum. It seems like there was a sign for Casa Cenote Hotel on the highway. The cenote is almost at the end of the road and behind the Casa Cenote Hotel. There's a beautiful beach there too and legend has it that manatees would congregate where the cenote dumps into the ocean.

 Xel-ha 

This theme park provides adventure activities, natural, ecological attractions and water activities like diving or snorkeling in creeks, lagoons, natural wells and ancient caves fed by subterranean rivers flowing to the Mexican Caribbean Sea of the Riviera Maya. I’ve heard reservations are recommended for swimming with the dolphins.

www.xel-ha.com

Xcaret 

This natural history oriented theme park offers a full day of activities from archeological sites to swimming with dolphins to floating down an underground river. They have a great night show and huge enclosed butterfly exhibit and a big aquarium.  We liked it a lot.

http://www.xcaret.com/

Whale Sharks:

I haven't done the whale shark trip but want to. The whale shark is the largest fish on the planet and grows up to 60'. They are peaceful plankton feeders. They congregate off Isla Holbox in June through September 15. You can swim with them, if you dare. One tour operator: http://www.holboxwhalesharktours.com/whalesharkvideos.shtml

Isla Mujeres:

We went to Isla Mujeres without a car pretty easily. Take the bus to the central bus station in Cancun. Go across the street to the bus stops in front of Commercial Mexicana. Here you can catch the R-1 or R-2 bus to Puerto Juarez/Gran Puerto. We ended up seeing a collective with it in the window and took that. Any bus should cost 6 pesos per person. Once at Puerto Juarez, it is 70 pesos round trip to Isla Mujeres.  Isla Mujeres is a lot like Cozumel only smaller. We really didnt see any special shopping. There is a grave marker of some pirate in the cemetary at the north end of the town. The south tip of the island is nice. You can climb the lighthouse and get a nice view of the sea and hotel zone. There is a zip line there over the ocean. The eastern side of the island was surprisingly rocky. We rented a golf cart for two hours which was enough for us. There was a nurse shark penned in a cage you could swim with at Playa....it was pretty depressing frankly. There appear to be a wide variety of dives you could do there and the cost was pretty cheap.

Hidden Worlds Cenote:

You see the brochures all over and it looks a little cheesy, but this turned out to be a very good snorkel tour. We took the two cenote tour for $40, which included a wetsuit and snorkel if you need it. Don't take fins as they are not allowed. Our group of 6 had a guide. The water was as clear as advertised 300' visibility they say and the stalactite and stalagmite formations were amazing. They take you out to the cenote in an old truck over a windy road. (could have done without that part) You enter the cenote through a fairly small hole and down a ladder. The underground area has underwater lighting and they give you flashlights. It was great. We really do recommend it. Link:   http://www.hiddenworlds.com.mx/

Isla Mujeres:

This is an easy day trip. Take the bus to the bus station in Cancun. Then a cab, collectivo or bus not far to Puerto Juarez. (R1 or R2 go there, so do many others, just ask the driver)  It was a nice half hour boat ride for 70 pesos round trip. The best way to see the island is to rent a golf cart. We got one for two hours ($300 pesos) and that was plenty of time. The south end of the island is very pretty. Be sure to climb the lighthouse. There is a zip line there with stations build out in the water.  Nice view of the hotel zone. The east side is quite rocky with spectacular surf. Be sure to check the return schedule. The lines can get long in the afternoon, and they did not seem to have extra boats. Diving trips there seem more reasonably priced. This is where the cave of sleeping sharks was discovered.

Isla Contoy

You can't stay on this island. It is a wildlife sanctuary. I don't know much about it.