You can get to the cayes by water taxi from Belize City, or a flight from either the international airport or the municipal airport.
By flight from the international airport:
No need for taxis. The flight to the cayes departs from the airport you arrive at. This is the easiest and quickest, but not the cheapest.
For the other options, you'll first have to get to Belize City. There are no buses from Philip S. W. Goldson International Airport (PGIA). A taxi to Belize City costs BZ$50.00 (US$25), and takes just under 30 minutes.
The municipal airport is in Belize City. Flights to the cayes might be cheaper than from international, but there is the added expense of the taxi fare.
By water taxi:
Ask the taxi driver to drop you at one of the two water taxi companies. If you don't specify which one, most taxis will probably go directly to the Caye Caulker Water Taxi at the Belize City Marine Terminal / Water Taxi Terminal.
For schedules and fares, here are the links to the 2 Belizean airlines and the 2 water taxi companies:
a) Maya Island Air, http://www.mayaregional.com/
b) Tropic Air, http://www.tropicair.com/
c) http://www.cayecaulkerwatertaxi.com/
d) http://www.sanpedrowatertaxi.com/index.html
As to which caye to go to, Ambergris Caye and Caye Caulker are different and the differences have been discussed here before. Try searching the forum for more opinions. Basically, Caye Caulker is more laid back, while Ambergris has more shopping, bars, restaurants, nightlife, etc. You can snorkel off AC and CC; other forum members will tell you which one is better.
For your other activities, San Ignacio would be the best base. Belize City is not the best place to stay. San Ignacio has a good selection of accommodation and restaurants, and you'll be close to all the tour companies.
Here's a site I started with bus & travel information:
http://belizebus.wordpress.com/