You've received a lot of respectable advice above. I'll throw in some other tidbits to keep you going towards your goal.
I watched the following documentary, with horror and amazement, and a bit of jealousy. I'm amazed they survived, horrified that someone would try to do this as they did and finally jealous that they completed the journey. Hold Fast, a documentary about anarchy and sailing - Boing Boing
1) To your question about size and cost of a boat to do such a journey; An opinion has some weight to it, but documented experience is a better source. Here is one source: The Bahamas - Sailboat
The boat documented, Suselle, just sold for around (or under?) $14,000.
Further to your question on size and cost, almost every sailor I know records all sailing / boat costs. You would do well to talk to some local sailors that are near your level of comfort and ask if you can discuss detailed costs. Not the off-the-cuff, over-breakfast costs, but hard numbers. That will give you a better idea than Internet opinions from people with varying comfort levels.
2) No one can answer your second question but you. We can help you with specific subjects and skillsets that you will need to learn. But your aptitudes and resources will dictate how fast you can learn and gain experience. I was talking to a friend of mine about the skills required to sail on a voyage. I jotted down on a napkin an outline or curriculum of subjects she would need to learn in order to take on a long voyage. Perhaps you have a mentor that can do the same for you.
With regards to your last comment about a "month of experience", well, let me rephrase it this way. If you were to learn to drive a car in a parking lot in Florida, and practice your skills in a parking lot in Florida for one month, would you be able to drive in the icy mountain roads in Europe? The answer is "maybe". If the weather is good and you drive without making mistakes, then you'll be fine. If you get caught in a blizzard or your attention waivers, your mistakes may end up with serious consequences. You gain experience by making and recovering from mistakes... one month doesn't give you much time to make those mistakes.
My last bit of information is about the common advice about "relationships" on a boat. My 'wife' and I have a typical relationship and we travelled 6 weeks in the Great Lakes on a tiny boat with our dog (same boat as Suselle above). We faced our good days, we experienced trying days. We fixed poopy head hoses, we swam in crystal clear crisp cold water. We were swarmed by stable flies, we were suntanned. We lived with one-burner alcohol stove and an ice box that rarely had ice, but ice makes beer that much better. We motored in calms and sailed in 14' waves... and at the end of the day, she's still willing to sail with me. If we can do it, you can do it... even though extreme's can happen, ordinary people will survive.
ok that wasn't the last bit. Check out Bika's blog(s) and see where a little budget and a little spirit can take you. They've been sailing since Norway 2005 Home - Bika