I think many of the posters before me made some great suggestions and price can vary wildly depending on what level of comfort/splurges you want to indulge in. For me a dirt cheap trip (from the east coast US) would be fly into Cancun, ADO bus to playa del carmen, dive Playa and the cenotes or catch the ferry to Coz. Or if I had more time then money go overland south into Belize and dive out of Ambergris or Caye Caulker or chicken bus it down to Placencia. For pure ease, Playa wins and can be done very cheap. Coz is beautiful but currents can be overwhelming sometimes, and definitely deeper dives in my experience.
I love the Caribbean, but have never been able to do it for less than double the cost of the Yucatan Peninsula. For me, the food is vastly more expensive, ditto for lodging, there is limited public transportation and dives cost more in my experience. Having said that, am going down to Curacao with the SO in December and from what I can tell seems like another good option for beginning divers, just pricier.
so suggestions for ease and infrastructure
Yucatan Peninsula-cheapest
Cozumel-cheap, great diving, some currents
Belize-mid level price, good barrier reefs, no beaches except placencia
BVI- expensive and polished operations
Caymen Islands- logistics for inter island travel, costly, very professional
More logistics but nice
Corn Islands- Nicaragua- will lose at least a full day in travel each way, minimal infrastructure, beautiful but needs planning, knowing Spanish very helpful in most of country, less so on Corn Islands themselves. (Personally really enjoyed Nicaragua)
Roatan or Utila-Honduras - multiple flights and logistic possibly an issue, variety of cost levels, good diving, Spanish very helpful
