When we got certified we were living in Washington State (eastern side) and didn't want to do training there because I hate cold water. We were on a two week Christmas vacation with family and arranged with a dive shop in Puerto Vallarta (Pacific Ocean) to do our classroom learning (this was before e-learning) and check out dives there. There was no pool work because there was no pool so all of our work was in the ocean. Since we knew we'd never go diving in WA, Mexico was a good place for us to certify. As I recall, the whole process took four days. A few years later we got our AOW in Cancun (by then we'd moving to AZ). Several years ago we did e-learning and got our Nitrox certification in AZ. There was about a three hour classroom session and no check out dives. We've never felt uncomfortable not certifying where we lived locally and have done a fair amount of diving in the tropics. Now that we live in Florida, we can dive, dive, dive.
We love Cozumel and have been diving there for years. Some of our earliest dives were there. Yes, the currents can be swift at times, but there are plenty of sites for new divers to feel comfortable. I would suggest, however, to hire a private DM for your first boat dives. That way you will have all the attention given to you, and the person will be able to help you get comfortable in the Cozumel waters. In Coz, there is supposed be one DM for every eight divers when the diving takes place in the marine park (which most of it does). So it'd be nice not to have to share a DM with the rest of the boat. There are many great dive shops in Coz. If you go to the Cozumel forum, you'll find tons of information to help you choose the right shop.
Welcome to the wonderful world of diving!