I've become very suspicious of any fan of The Beginning of Infinity (BoI), who says it's great, but who has not read The Fabric of Reality (FoR). I've seen a repeated pattern where people who haven't read FoR are shallow fans of BoI who don't know much about Critical Rationalism (CR). Overall, BoI is more popular than FoR, but is attracting worse fans than FoR did. Anyone who has read both and likes BoI way more is also highly suspicious and likely didn't understand much from either book. I don't think "BoI is way better than FoR" is a reasonable opinion. Anyone who goes around recommending BoI, but who recommends FoR much less or not at all, is probably a bad thinker.
Also, Deutsch's books should be read visually (which makes it a lot easier to catch more details, take your time and only advance to the next paragraph when you're ready, reread things, etc.). Be very suspicious of the understanding of anyone who's read them only as audio books. (Deutsch is irresponsibly selling audio books with no warning that understanding his ideas from an audio book is unrealistic. It's an unsuitable format for a first reading of his difficult, wordy books that contain many long, convoluted sentences. Audio books are fine for a casual second reading to review the book a bit while knowing you're missing a lot. They're also fine for a blind person who is very experienced with audio books, listens at a much slower speed than they usually listen to books at, and regularly rewinds to hear parts again. But a sighted person who starts with the audio book is almost certainly fooling themselves rather than actually understanding much.)
BoI is doing a much better job than FoR of attracting social climbers who talk about the book as a way of bragging. BoI is more popular, and it's a bit easier than FoR to read in a shallow way and think you liked it without learning much. BoI also has more things that can be used as slogans or sound bites.
If you haven't read either book, FoR should be read before BoI. I strongly recommend reading them in the order they were written. FoR does a better job of introducing and explaining CR ideas for new readers. FoR also does a much better job at introducing the many worlds interpretation of quantum physics. Deutsch put his most important things to say in his first book and didn't repeat them all in his second book (which would be fine, except that he doesn't tell anyone to read the books in order).
FoR is a deeper book with more technical details. It goes into more depth on some specific topics rather than focusing as much as BoI on being of general interest. It is a popular science book meant for the general reader, but some sections are less useful for most readers. In particular, the two chapters (11 and 12) about the physics of time can be skipped. The last chapter (14), about the end of the universe, is also skippable, especially the omega point discussion.
FoR talks about four strands, CR, evolution, quantum physics and computation. The key chapters in FoR to learn about CR and evolution are 1, 3-4 and 7-8. The CR and evolution strands are the more useful and easier to understand for almost everyone.
But if you're trying to learn about philosophy, I don't recommend starting with Deutsch's books. I used to recommend them more, but most people find them too difficult to learn anything substantial from, especially as a starting place. Most people who like them, and think they're learning something, didn't actually understand much.
Some good places to start learning are my Critical Fallibilism website and Eli Goldratt's books (especially The Goal, It's Not Luck, and The Choice). After that, you'd have a better chance to actually learn from FoR, though I'd recommend first reading chapters 1 and 2 of Philosophy: Who Needs It by Ayn Rand and reading Understanding Objectivism by Leonard Peikoff (which talks about how to learn a philosophy).
Also, if you don't already read books regularly, you may be more successful by first getting into reading and then trying to read FoR later when reading a book is already a common, easy and enjoyable activity for you. Many people should start by trying to form a habit of reading regularly, and enjoying it, using fun books like novels. I like Robert Heinlein best for sci-fi (start with his juveniles) and Brandon Sanderson for fantasy. My reading recommendations in the previous paragraph are much easier to read than Deutsch's books, and might actually work for newer readers, though they're significantly harder reading than Harry Potter. Also, you may want to start getting into reading more with audio books or text to speech, and that's fine and works well for many people, but at some point you should transition to also getting comfortable with visual reading, which you'll need for reading authors like Deutsch or Popper.