For Staying: Hotel Cozumel. Pay for the all inclusive package (room, airport transfers, food, and drinks). I have always had great experiences here. Clean rooms, friendly staff, great food and drinks. The pool is nice. There is an under the street walk way to the other side of the hotel (on the beach). They have their lunch buffet and beach bar there. You can also find their dive shop, boat pier, dive lockers, dive gear cleaning station, and stairs into the ocean (for shore diving) there.
We usually rent a tank and weights the night we arrive for some shore diving to check our buoyancy (for dive boat weight) and to get a nice 20ft shallow dive with some yellow stingrays, parrot fish, and various other beautiful sea life.
For Diving: I like Dive With Martin. Most dive shops will pick you up at the Hotel Cozumel pier. You can also add diving to your package and dive with the boat / shop from the hotel. I have tried 4 different dive boats in Cozumel and never had a bad experience just a better one with Dive with Martin.
I recommend calling ahead to make dive boat reservations and most boats will meet you at the peir outside the hotel. Good luck.