Congrats on being hooked on the best sport on the planet!!!
Cozumel and the Cayman Islands are both beautiful places to dive and I have enjoyed them several times. I have not dove with Blue XT but those who have commented have had good things to say so far so I have no reason to believe that they would not be a great operation to explore.
I am going to respond in a slightly different fashion from those who have posted above (I think).
Since you are planning the trip for May of 2010, you have plenty of time to get certified before you go. I would suggest, as others have noted, that you get certified BEFORE you go on your trip. Take the time between now and then to work on your diving skills, comfort in the water, and proficiency. By the time May rolls around, you should hopefully feel comfortable enough to dive in the Caribbean and take in all that it has to offer.
Personally, I would not want to spend my time in paradise learning to dive. I want to spend the time in paradise actually diving. I had a former student send his wife to Cozumel to do her check out dives and she could not get past the mask skill. She never completed her certification and spent her time in paradise on dry land!! If you are going to have issues or unforeseen problems diving, then I would suggest that they be worked out before you get there. Otherwise you could be wasting time in paradise (not diving that is).
Additionally, diving in Cozumel especially usually involves diving in current. This is a very different experience for new (and seasoned) divers who are not used to it. It can be confusing and disorienting and may take a little getting used to. If you are comfortable with your diving skills before you get there then the transition may be easier. Caymans has a little current but I did not experience much when I was there.
You have several options to choose from and the posters here I think have given great advice. My 2 cents would be to get certified before you go and spend the extra time honing your diving skills. It will make diving in a new environment more enjoyable.