You have some excellent advice above; just to reinforce some points: Yes, the plan sounds very cool and doable! Learn all you can first; after sailing lessons, a week chartering/hiring boat and skipper will teach you a lot. Most sailing clubs hold informal "beer can" races over summer - use them to learn not only about sailing, but about different boat designs, their advantages and disadvantages. My concern it that you take the time to (a) make sure you enjoy this life, and (b) build up enough knowledge when it comes to buying the boat - which is a big decision. In fact many people buy a small boat - e.g. a Santana 22 - to build experience before the big purchase. Building up a base level of knowledge may take a while, as people have pointed out.
My other point is regarding the budget. The good news: $20K should be enough to buy you a nice *coastal cruiser*. You'll need a survey, probably bottom job, minor repairs, perhaps taxes...so if your total budget is $20K, your boat price is likely $15K (many variables!), but at that level it is still doable Your coastal cruiser should be fine for hugging the US coast, and around the Caribbean - but probably won't be up to "blue water", global cruising. Blue water boats tend to cost a lot more.