Islandlove--Does, by chance, your cousin who lives in Jamaica have space where she lives to host the other two of you to stay for two weeks? If she also lives not far from a beach, that might be the most economical way for the three of you to vacation together.
Regardless of what tropical paradise you choose to visit, you'll need to each spend a good deal flying there. You'll also probably encounter a trade off between how low your accommodation costs will be in particular destinations and how much you'll need to pay to get there. Fortunately, the three of you will be able to reduce your hotel costs considerably by lodging together. Four possibilities to consider:
(1) The Florida Panhandle--AKA "Redneck Riviera" (Fly into Panama City, rent a small car, maybe rent beach place by the week; No, its not the cheapest place in the world to stay, but it might be quite practical if you find favorable airfares.)
(2) South Florida (Fly into Miami or Fort Lauderdale; Hotel lodging and meals might cost more than for #1, but without the necessity of your renting a car for two weeks. I believe there are some youth hostels at Miami's South Beach.
(3) Mexico's Caribbean Coast--AKA "The Riviera Maya" (Fly into Cancun and then base yourselves in Playa del Carmen, Puerto Morelos, and/or Isla Mujeres; Cheaper food and lodging than in the USA, but probably more airfare to arrive.
(4) Honduras--(Fly into San Pedro Sula and then bus to the Caribbean. Higher air cost getting there might be partially compensated for with lower lodging and food costs, compared to #1, #2, and #3.
Final Comment--Be wary of attaining what you seek. Each option above is not merely "warm" but extremely hot and humid during the summer. Following a Canadian winter you now seek heat but when immersed in it your perspective may change. With limited funds consider flying instead to somewhere in the southeastern USA, such as North Carolina or Virginia, with access to rafting, cool green mountains and also (a day's drive away) ocean beaches. Where ever you go enjoy your time together!