This is a big question. As Digital Dracula says, it's like asking, "what car should I buy?" or "How do I write music?" But to give you a starting point:
It's pretty easy to self-publish these days, so you can be confidant that you can do it. Therefore, the first task is to actually write the book. When you're close to getting done, then is the time to get into the details of self-publishing. But it certainly helps to have an idea where you're going.
You can publish a printed, paper book, and/or an electronic book like a Kindle or a Nook or a simple PDF.
For a paper book, you contact a self-publishing company. There are basically two kinds:
(a) The ones who will print just about anything you give them. You send them the text of the book in an appropriate electronic format -- usually PDF -- and they print it. You have to arrange for any proof-reading, format the book exactly like you want it, design the cover, etc. They will then make it available for sale on on-line sellers, like Amazon or Barnes & Noble. Any marketing beyond just showing up when potential customers search is up to you.
(b) The ones who proof-read your book, do the formatting, design the cover, etc. Again, they will then make it available for sale.
Group (a) these days charge minimal fees up front. There may be a setup fee or a cost to get proof copies printed, but typically less than $100. Group (b) charge considerable amounts of money for their services, typically thousands of dollars.
In either case, you collect royalties when copies of the book are sold.
Do you live in the U.S.? Your profile mentions the U.S. but doesn't say if you actually live here. But here in the U.S., I think the main "do it yourself" publishers are CreateSpace (which is owned by Amazon), Lulu, and Lightning Source. You might want to check into them. I published my first book with Lulu and my second and third with CreateSpace. Both of those are very much geared to working with individual, self-publishing authors. Lightning Source primarily works with big publishing companies. They will work with an individual author, but that's not their target customer.
I don't want to name type (b) publishers because I've never worked with any of them. I'll warn you, many of them are borderline scams: they charge you large amounts of money and give very little in return. But some are legitimate, and if you need the services they provide can be worth the investment.
If you want to publish an electronic book, you can work with Kindle Digital Publishing. They are geared to working with individual authors. I've published two of my books through them. They'll take you through the entire process of creating and publishing your e-book. I understand there are other services out there -- I've heard SmashWords brought up a lot -- but I've never used them so I can't say much about them.
Expect most or all of your sales to be on-line. It is very difficult to get a self-published book by an author who is not already famous into bookstores. I've talked to a lot of writers who have devoted considerable effort to getting their books into bookstores, and most got nowhere. The few success stories I've heard are people who got their book into some small local bookshop. Your chances of being picked up by a national chain are minimal. Personally, I don't waste my time. I wrote a computer book that is sold in a college bookstore, where the college is using it as a textbook. I have a book about the Bible that is sold in a tiny bookstore attached to a church. That's nice, but probably 70% of my sales are through Amazon.