Since you said you liked the DOs too, here's a few things I've picked up from here and from a few very experienced divers I know when I was new (still am, really). Keep in mind that all of my dives have been drift dives done exclusively in Cozumel. Some of these things might not necessarily apply to all dive locations.
Do look at your SPG when testing your air before you jump in the water (BWRAF). If the needle moves as you breathe in, your air is probably not turned on properly. You might not notice it on the surface, but as you descend this could become a problem. Tell your buddy about this check as well. Most instabuddies I've dove with didn't know about this.
Do pull your mask down and wear it around your neck while getting back on the boat (vice putting it on your forehead like sunglasses). If a wave smacks you in the face, you don't want to lose your mask because of something so easily avoidable.
Do remove your fins last before boarding the boat. If you're hanging onto the ladder and removing gear, and if you lose your grip for some reason and get separated from the boat, you better have your fins on so you can swim back to it. You are no match for the ocean without them.
And some things from me:
Do obtain and learn how to use an SMB ("safety sausage") as soon as possible. I never thought I'd need one until I blew through my air in 36 minutes and ended up surfacing before everyone else. Do have someone train you on how to use it properly. They may look easy enough to use, but they still take both proper instruction as well as practice.
Do obtain a whistle and attach it to your BCD. If you surface too far from your boat, you might need to use it to get their attention or the attention of another nearby boat. You should also know the name of the boat you are diving from as well as the channel they operate on. This information should be given to you during the dive briefing. This is information that you can provide to any other boat in the area so that they can radio your location to your boat.
Do hydrate well on the days you dive. Every morning I woke up to dive, I would drink water until my pee was clear, and then I'd drink a little more. Then I'd drink some on the boat. Hydration is an important part of helping to prevent DCS. Watch what you eat, too. I read something on here in the Accidents and Incidents section about someone who "took an undeserved hit" (got bent - DCS - unexplicably). They later found that the packaging on one of the foods they had been recently eating mentioned something about "increases nitrogen loading/intake" or something to that effect. I think this person may have also been working out at the gym after their dives too. I know I read something along those lines.
Do read as many threads in the Accidents and Incidents section of this forum as you can stand. I promise you will learn things never taught to you in your training, or even things specific to certain dive locations. Had I never learned about downcurrents (also called downdrafts or downwellings, apparently), I don't know what I would have done if I was ever caught off guard by one. I was surprised to learn of their occurrences in Cozumel, and despite not having seen one, I know there was one at least a few months ago. Thankfully everyone was getting on the boat when it happened. I have no idea at what frequency they occur there.
Do pick up the book "Diver Down". It's an easy read, based on analysis of some dive accidents.
Do ask lots of questions, and ask the same questions from different people. I've asked multiple folks, "what would you do if you were at depth, and while putting a little air into your BCD for buoyancy control purposes, the inflator button gets stuck, rapidly filling your BCD with air?" I had something similar happen to me when I was getting my OW, but thankfully I was already at the surface. I don't know a single person who went through the PADI course that was taught about disconnecting the inflator hose while underwater if the need ever came up. It might sound like common sense, but things can happen so fast underwater that if you've not practiced a certain skill before, who knows how you will react when an incident actually occurs. Know where all your dump valves on your BCD are as well.
Diving feels deceptively safe, but in reality it is inherently dangerous. Being aware of the dangers and practicing your skills to mitigate them could help to save your life, or the life of someone else around you. Stay safe, and I hope all of your dives are enjoyable ones!