1. Finish up your certification at home. You really don't want to spend time in classroom or pool work during your vacation, and while a referral certification (where you do classroom and pool work at home, with a referral letter to a Cozumel Instructor to finish up your certification dives), you've got lots of time, get your C-Cards and do a little weekend diving at home to hone your skills.
2. Minimum equipment needs are mask, fins, snorkel, and protection - wetsuit, skins, skins plus, depending on your sensitivity to cold. I personally have never needed more than a "skins plus", a Henderson dive suit which is fleece lined and has 0.5 mil of Neoprene, but I carry a lot of my own insulation with me (the same as used by whales). Water temps in Cozumel during June and July will be warm - at least 80, more likely 82-84 degrees. But if you need more warmth because you get cold easily, or you are going to be diving every day (multiple days of multi- tank diving will result in a cumulative loss of core body heat), a 2-mil wetsuit should be plenty. Snorkel is not really essential, and though I was trained to always have a snorkel with me, I no longer even use one diving. But I like to snorkel on off-days and afternoons, so get one. I didn't add computer to essential equipment (maybe only because I started diving so many years ago when dive computers were very rare and very expensive), but they are very good to have. Once you decide that you are definitely going to be divers, get a computer, and get one that's Nitrox capable. I now take two computers with me, my main computer and a backup.
3. Go with a smaller Dive Op that will provide you personalized service - there are dozens out there, including Aldora (the biggest of the "personalized" dive ops, and a favorite of many because of Steel tanks, very long bottom times, long surface intervals at a beach club, etc.), Blue XTSea, Tres Pelicanos, Dive with Allison, and others. Also, consider hiring a private Dive Master to accompany you on your first day - it will really reduce the nerves that will accompany your first ocean dives.
4. Favorite dive spots -- pretty much all of them! They all have something different to offer, some with magnificent coral structure, swim throughs, a flight along a wall watching the abyss below you (wait until you have at least 3-4 days of diving behind you before you try the walls). If you are going to do 8-10 days of diving, don't worry about it, your Dive Op will pick sites that are appropriate for your skills and experience. Cozumel has lots of great spots that are perfect for Newbies, and a few (Punta Sur, Devil's Throat, Maracaibo, Barracuda) that require quite a bit more in skills and experience.
5. In June and July, you will almost certainly see rain, don't worry about it. Even if it rains every day, it won't last long. You MIGHT get a Norte, when the wind comes in strong from the North, sometimes strong enough to close the Port and cancel diving, but these are rare, so don't worry about them. July WILL be hot and muggy - you DEFINITELY want a Hotel with good AC in the rooms and a decent Pool. There are downtown hotels that are nice (like Casa Mexicana) that I won't stay at in July because the pool isn't big enough for me. Also, plan for Mosquitos, bring repellant.
6. General tips. One of the reasons I love Cozumel and keep coming back is that it has very friendly People and a wealth of great restaurants, most of them downtown. Because of that, I don't do All-Inclusive resorts on Cozumel, and recommend against them. Maybe, as Honeymooners, you want to be isolated on a big resort, and maybe you aren't big foodies so the monotony of the same food over and over at an AI resort won't bother you. But unless you make a few trips into town, you'll miss a lot of the charm of Cozumel, and if you stay at the far southern resorts, taxi trips into town will be expensive.
Enjoy your trip.
---------- Post added July 3rd, 2014 at 09:50 PM ----------
Ps -- on diving Cozumel. As noted, Cozumel is all drift diving. You drop off the boat, descend to depth, and let the current carry you along. The group stays together behind the Dive Master, ascends as a group, and the boat picks you up. It is the easiest, most relaxing diving I know of. I've gotten lazy in my old age, and I don't regret the fact that I haven't looked at my Compass or tried to navigate under water in at least 5 years. BUT some divers get nervous or overworked in drift diving because they don't follow the simple rules. First, DO NOT FIGHT THE CURRENT. It is your friend, doing most of the heavy lifting of moving you along the reef. Second, STAY UP-CURRENT FROM THE DIVE MASTER. If he stops to look at something, or point out something to the group, stop with him. And try not to form a tight crowd around him if he's showing you something small, like a Toadfish hiding in a little hole - everyone in the group will get their turn. Third, CONTROL YOUR BUOYANCY AND STAY WITH THE GROUP. When drift diving in a current, the water can move at different speeds at different depths; if the DM and the group are all at 85' and you decide to stay at 60', you might well find that you are being swept along faster than the group and get out ahead of the group. Trying to swim back against the current because you've drifted out 80 or 100 feet ahead of the DM can make drift diving harder work than it needs to be. Fourth, learn the tricks to minimize the effect of the current or avoid the current. Stay horizontal presenting a small cross-section to the current. Get close to the bottom where the current is usually weaker. Duck behind a coral head to let it block the current. Stick a finger or two into sandy bottom to anchor yourself (PLEASE don't grab the coral, we don't want you killing it, and with the wrong kind of coral, you'll be very sorry!). Most of the time the current in Cozumel is fairly gentle, but if it kicks up a bit or goes in crazy directions, don't worry. Just follow the rules and enjoy the dives.