For Cozumel, and I've been there many times and dived with several providers - my nephew even worked for one as an instructor, as we are all divers in the family - I can recommend one provider especially highly:

Aldora Divers

Trip Advisor reviews of Aldora Divers

Honcho man Dave Dillehay's spacious Villa Aldora is the best place to stay (north side of the island so you can duck into town and avoid the huge cruise ship crowds, wide variety of accommodations, boat picks you up there).

Aldora has an all instructor staff (no divemasters, one instructor per fast six passenger "six pack" boat), they set you up with steel tanks of various sizes to equalize everyone's breathing, have a superb instructional staff, and with their boats get you to the best places while there's nobody else on them.

Lady JDiver did the pool and bookwork at home, did her certification - and some hundred plus dives - with Aldora. As a rescue and master diver I can say these guys are superb; Manager Guillermo "Memo" Mendoza runs a tight ship that is very customer oriented, and is a superb critter spotter.

They may look a bit more expensive, but then again, I've spent 1, 1.5 and even 2 hours underwater with them, and you will get more bottom time than you will with the other guys, they match divers up for skills, and can't be beat IMO.

(I've got 58 years of diving in 32 countries, certified wreck, rescue, master, etc. for reference.)