OK, advice purely on sailing:
Take a series of courses at an ASA-certified sailing school, or one of the well-known independent schools. Basic, advanced/spinnaker/racing, liveaboard/bareboat, then passagemaking. The first two will typically be three days, the latter two week-long each. See how you take to it, you can schedule them all within a two-month window if you want to push, or spread them out further. But, I would urge you to get at least into the "bareboat" course and then look into crewing or charter opportunities. The more you can schedule during bad weather--the better. Because going around the world, you will have to do foul weather sailing and that really can put a damper on things.
Experiment with seasickness meds in adavance, at home, so if there is a bad reaction you have the luxury of being warm and dry at home when it happens. If you are one of those lucky people who is never seasick, congrats!
There are plenty of books and web threads on what is a "bluewater" boat, you may want to really tear into them and make some personal decisions about how much boat you want and which philosophies you want to follow.
Once you've taken the basic courses, become a dock rat. Find some local yacht clubs, find out when the race nights are, and ask who needs crew. With boat shoes, adequate clothes, and a smile, you can get picked up as crew almost every night. That's a great way to learn what people are doing, why they are doing it, and get different opinions. Racing skills can be very important to a cruiser--you may NEED to get your boat moving faster sooner or later, and over the course of a week or two-week passage, that little biut faster can make abig difference. It also forces you to learn concentration.
Also consider that if you are going round the world solo, you will be in violation of the laws of every nation. Yes, it is done all the time, but going solo means that you cannot keep an adequate watch on deck at all times--which is a real danger, not just a legal violation. Picking up partners for different segments of the trip might be a way to add "local interest" and raise more support, as well as increasing your safety.
Frankly, I think you'd be safer, and raise more support and interest, by looking at something like "two pros, plus two kids in remission on each segment" so that the kids were being given the opportunity to be part of things, also ensuring 2 crew on watch at all times.