What you're saying isn't impossible
. There are ways to go about it and ways to throw money away, not to mention get in to trouble. For starters i would say don't be one of those people who disregards just how little our oceans care about you. A great number of people have lost their lives foolheartedly chasing their dreams of sailing the world. Now that the scary part of sailing is said...
Start with a good boat. Make a short list and wait poised like a cat to pounce on a good one because you...learned, read, then read some more. Not only about what makes a good boat, what keeps a boat maintained, how to manage a cruise, navigate, work a radio, and of course how to keep her afloat. It's estimated you need 1000 hours to become competent at anything, 100,000 hours to master it.
Be prepared to pay. Someone gave me a pretty good estimate, key word being estimate that you can spend roughly half of the value of your boat fixing her up. Buying a boat that's fixed up may be your gig, but other than the odd deal now and again, you're looking at paying more and spending less on gear, etc. Weighing those variables and foreseeing all those hidden expenses becomes nearly impossible. I can attest with a stack of West Marine receipts just how hidden those costs can be.
Then i would have to suggest taking a good look at avoiding going solo. Singlehanding is arguably extremely dangerous, especially for someone who could stand to log some miles. Way more than having crew or at least another hand. You may just want to start crewing. Either that or like you said, hit up a school and spend some time in classes. Either way jumping out there without the knowledge to get you there safely is pretty scary stuff.
As for me..it's awesome our similarities. I am 38, male to be exact, and i am pretty much in your shoes several months, if not years later. I have just purchased a 1978 Morgan 382 and am in the process of refitting her and getting her ready to cruise. She's quite a bit bigger than your idea but it's what i figured would keep me sailing for the long-haul without giving up too many comforts. I waited...& when the time was right i paid $15k for her and she's a gem.
All this stuff is just what i've learned over the last 6 years getting ready. I would say go hang out at the docks, yacht clubs, and sailing events until you catch a groove. You will most certainly find one as people in the sailing community are usually truly helpful people.