1) Of the books you buy for yourself, pick your favorites and review them. Submit those reviews anywhere that will take them; what matters at first is building publication credits, rather than making money. Whenever you get a review published, send a copy of it to the publisher (and author if you have their address) of that book; this is called a "tearsheet" (because before copy machines and internet, it involved tearing a page out of a magazine). Note that tearsheets used to be customary; now they are not. So if you send tearsheets, publishers and authors will love you, and give you oodles of books, and tell all their friends about you.
2) Once you have some published reviews, watch for regular reviewing gigs to open; then apply to them, including some of your tearsheets as an examples of your skill. The first one or few you get will probably be volunteer. However, many review markets provide you with free review copies to read. This saves you money because you don't have to buy all your own books now.
If you can't find author contact info directly, watch for people on the "attending guests" list at conventions. You just print off tearsheets of your reviews for anyone who's at a con you're attending, and catch them after a panel, signing, or other activity they're scheduled to do. It's a way to make a lot of friends very quickly, because Feedback Is Candy. This trick also works for publisher parties at conventions, but it's easier to find publisher contact info: theirs is usually printed inside the book.
3) At this point, you can send some tearsheets to a favorite publisher and say, "I am a reviewer with a column in (market name). You may have seen my reviews in other markets such as (list). I have enclosed some tearsheets so that you can see my work. Please add me to your list of reviewers. You can send catalogs or review copies to me at (address)." If you want to get fancy, you can specify whether you want just final ("shelf") copies or will also review from Advance Reading Copies (ARCs), galleys, manuscript pages, electronic files, etc. Some publishers also save their "seconds" (shelf copies with a bent cover or other minor flaw) as review copies rather than destroying them; this custom saves trees and is to be encouraged.
If you've made friends with authors, you can also ask them to put you on the list of people to whom their publisher will send copies of their new books coming out. Most publishers have a publicity form for authors to fill out, and it has a space for "please send a review copy to these people." This can add up to a nontrivial amount of free books, for which consideration you provide reviews and tearsheets.
4) Now that you have some good publication credits built up, you can hunt seriously for paying markets. It's easier to break into those if you've written a goodly number of reviews already. Most review markets pay modestly ($10-20/review is common) but if you already enjoy reading, this is a pretty good deal when you count the free books and the fact that reviews are short and only take half an hour or so to write. Some review markets pay quite a lot more than that -- I've seen listings for $100 and up, though I've never managed to get into those markets.