You can fly to almost anywhere in Central America from Miami, including Belize City. Miami is one of American Airlines' large hubs; you won't find nearly as many flights to Central American destinations from other FL airports.
In addition to the experience question = How much diving do you want....is diving is the primary focus (4-5 dives per day) or do you want a more leisurely pace (1-2/day) with other activities thrown in? Do you want to combine the diving with other activities/excurions on land (Mayan ruins, ziplining, river tubing, etc.)? Are pretty beaches important? Are you happy with basic/rustic accommodations or do you want 5 star? Are the teens younger where an AI would be best or 18-19 and want nightlife? Would they be happy on a small island/caye 30 miles offshore with not much to do besides dive, snorkel or read in a hammock? Are you 100% set on Belize and all it has to offer....or are you open to other areas of the Caribbean?
Some very basic info on Belize diving destination options (I've planned 2 trips for our family of 4 there recently...so not an expert, but some experience with the offerings):
- Ambergris Caye = relatively large offshore island that is very touristy and developed, albeit on a small scale. The main town (San Pedro) has many restaurants, bars and small resorts. We were there for Christmas week 2014 = my teens (18 and 20 at the time) enjoyed it because of the "action" (we stayed right in the main town of San Pedro at a small hotel during a trip last year). Farther out of town are various condo developments in various price ranges. Many dive operators. I found the dive quality to be decent and enjoyable....but it will not blow you away depending on your dive travel experience. There are many dive sites in a 10-15 minute boat ride radius. There are options for daytrips to other small islands (Caye Caulker) or Mayan ruins
- Southern Belize Mainland/Placencia area = quieter, some very nice resorts with integrated dive ops. Logistics are easier if combining land-based activities with diving/water activities. Boat rides to better dive sites tend to be somewhat longer since you are on the mainland.
- Offshore keys = this is where the best diving is (other than via liveaboard boat), but these resorts are further offshore and they are small and pretty quiet (we are trying one of these in 2016). Boat rides to great diving are 5-10 minutes. If kids are happy diving, snorkeling, kayaking, paddle boarding, etc., could be a great option....but there won't be much in the way of high tech fancy activities to keep them entertained (likely no TV or internet).
It's also possible to combine at least some of these with time in western Belize for some "jungle" experiences.
If you narrow down your choices to one of these, folks here can help with more specific resort recommendations.
Hope this helps.
Chris