I'll chime in here a bit. I think Alan summarized the connections nicely. I've taken the Q33 bus - E train combo to get from LGA to Jamacia, and I've taken the Airtrain from Jamacia to JFK (although never all in one go).
If you attempt the Q33 - Etrain connection, plan to spend at least an hour on that leg. The E-train ride is longer than it looks and it takes about 20-25 minutes. I've taken the bus on two occasions. Once it took 5 minutes; once it took 15. If you add in the time you'll spend waiting to board the bus and the train, you're talking an hour to make that connection. Also, if you're trying to find the schedule for the Q33, note that it is NOT an MTA bus, but is operated by a private company. Do a google search for 'Q33' and 'LGA' and you'll find it. During off hours, this is actually a pretty straight forward and easy connection to make. I think it is what alot of LGA and airline employees use because I've seen lots of them on it.
I'm not certain that the airport shuttle bus that Alan mentioned is still running between LGA and Jamacia. I know they used to, but I haven't seen that option listed on their website ever since the Airtrain started operation (they used to run to JFK too). Perhaps the resulting loss of business was enough for them to drop Jamacia entirely.
If you're really strapped for time, you can always take a cab from LGA to Jamacia. If traffic is light, it should take about 20 minutes and cost between $20-30. I actually did this at the end of last December and the cab driver got me there in 15 minutes - I tipped him accordingly. If you have to go with this option (due to a lack of time - ie late flight), remember that there is more than one cab stand at LGA. The one centered outside of the main terminal (where AA flies) is probably going to be packed with a 10-20 minute wait. If you walk down to the far end of that terminal (the exit end, not the entrance end), there is another stand that may be completely empty.
As for Airtrain costing a lot, I've used it several times and am readily willing to pay $5 for the easy connection. When compared with the previously available options (airport bus, cab, drive and park) it is the fastest, easiest, and cheapest way to get to JFK. At JFK, AA flies out of Terminal 8 and 9 (I think 9 is domestic). The travel time from Jamacia to terminal 8/9 (same stop) is 10-15 minutes.