I was born & raised on a boat so it is hard for completely put myself in your shoes (notably, I do not get sea sick, no matter what), but I personally think it is do-able. I have not done it yet, but I've sailed quite a bit and someday I want to start crossing oceans. Financially it isn't as bad as most people think. I actually think that the oppurtunity cost of quitting your job for a year or two is the biggest expense.
First off, you need to know how to sail at its most basic level. The cheapest way to learn is to get a dinghy (a Laser, maybe a Hobie 16 or Hobie 17 are all easy to find. Something decently high performance - you want to wipe out sometimes so that you learn.) You need a lot of time on the water and there is absolutely no substitute. You need to learn how to drive by the feel of the wind on your neck, and you need to learn which clouds are safe and which ones are going to wreck you.
Then you should step up to 30-40 foot keelboats. You probably don't want to buy one straight away. Learn on someone else's boat as crew for racing or deliveries. (a delivery is generally moving a couple hundred miles for the owner, i.e. the owner needs the boat moved from NYC to Maryland, but can't do it himself. He hires you to move it and pays all expenses. I've done this several times and it always seems to lead to the hairest situations because you are forced by the schedule to sail even if the weather doesn't cooperate. Generally you will be shorthanded 2-3 people which is good practice for offshore work)
So at this point you probably have been sailing for 3-5 years or so, know how to do everything on someone elses boat, and you need to buy your own boat. There is a lot of discussion on a fast boat vs a slow boat. A fast boat means you spend less time out there which is good mentally, but they are more physically demanding (bigger sails, crash through waves harder, get wetter, etc), but you can also avoid weather if you know a storm is coming. Faster boats break more often. A slow plodding boat can survive everything, but you can't avoid anything. Slow boats are cheaper.
You also need to figure out your route. Sailing non-stop around the world is one thing, but honestly that is kind of boring. I know 3 couples who have done it, and they made a point of stopping a lot along the way so it took about 3 years each. (1 couple stopped in New Zealand and taught school for a year, 1 couple was in Indonesia during the tsunamai on their boat in the harbor, and in the aftermath stayed for a year because they were doctors). The stops sound like more of the journey than the sailing honestly.
If I were to think of places to start, I'd subscribe to some sailing magazines. Cruising World comes to mind, there is also Sail, and Sailing. For websites, try Sailing Anarchy (rough crowd in the forums for sure, but there are some gems). You will definitely find them ridiculing people for what not to do on Sailing Anarchy.
Disclaimer: sailing is very dangerous. Way more dangerous than most people realize. I personally know 5 people that have been killed sailing in the last 4 years. It can happen in a lot of different ways. And I personally have been involved in some really close calls. Sinking boats, out of control boats, overturned boats. The fastest boat in the world (Rambler 100) just flipped and almost killed its crew off of Ireland. In the 70s the Fastnet race killed 50 people. The Volvo Ocean Race had a professional get killed in its last go-round. The ocean is no less forgiving now than it was in the 1700s. The Coast Guard can only reach so far, and when you are in the middle of the ocean you are absolutely alone.
As just one example, a few weeks ago we went out racing on a wednesday night (3 man high performance boat). It was a clear, calm night. Thunderstorms were rolling in, we saw them, we thought we could keep the sails up about 2 minutes longer than we should have. We knocked the boat over on it's side, the guy driving fell below and cut open his shin and was bleeding everywhere trying to get back up, the boat had no driver, and the 2nd guy on the boat froze like a deer in the headlights. Our sails were dragging us down and the wind was up like crazy and the rain rolled in to the point where we no longer could see our competitors who were 75 feet away. I crawled up to the bow and cut down one of the sails and hand over handed the thing into the boat, and when we finally got it all sorted out and the rain cut back, we were still sailing at full speed and realized we were in the middle of a reef. We got out of the reef without sinking the boat and decided to just quit the race and go home. I think my adrenalin was still going 4 hours afterwards.
I flipped my race boat a couple years ago while teaching my girlfriend to sail and the keel broke. The boat started sinking, and she started getting hypothermia. Another guy who was out that day coaching jumped in the water in full clothes to help me get it back upright and took her in by powerboat while I sailed the sinking boat back to the crane to get it pulled out.
The point is, unlike other sports when shit hit the fan, there is no time out. You don't blow the whistle and talk about it. If there are rocks coming up and your boat is out of control - you are going to hit the rocks and sink. If the boat flips - you better hold your breath and start swimming and hope it comes back upright. If something snaps and the rig falls down, better start cutting before it punches a hole in the boat and sinks you. When things start going wrong you can feel really exposed out there, with little to no safety net, so you better be prepared.
Anyone can do it but you have to do your homework. You have to put in the practice and preparation.
edit: My god, I posted that before reading the article itself, I can't believe how close his story is to my experience. Aside from the 90 foot wave comment (90 foot waves simply are not common, hurricane irene was throwing legit 30-40 footers around but it's hard to believe much more than that), he is spot on. I've never put a word on it, but initiative is exactly right. That day I went out and I mentioned the 2nd crew had the deer in the headlights look, he had 10x the experience of me, owned a boat bigger than mine, but in that moment he was completely stuck and useless until we started yelling at him what to do. This is an amazing account of what it is actually like. Really not that much hyperbole (which believe me, sailors are prone to, but it really is like this).