I’d start with Genesis (you can skip the genealogies), and then read John and Luke: John first, then Luke, and then Acts (I don’t like splitting up Luke-Acts because they’re actually one book, and should be read as such). After you’re done with that, come back for more suggestions. Daniel should be the last book you read in the OT, and Revelation should be the last book you read in the entire Bible.
Luke and John are quite different and focus on different aspects of Jesus and his teaching, unlike Luke, Matthew, and Mark (which focus on the same aspects). Genesis is of the world from the beginning, and is important for the understanding of John. Acts continues where Luke leaves off, and is a history of the early Church (Luke himself, like the author of Maccabees, is an excellent Hellenistic historian with a great care for the truth in history).
Genesis, John, Luke, and Acts are also very interesting books in their own right: they’re not boring or “Biblish”: they’re engaging just as a normal book is (and they’re all relatively short: the four books together, I doubt are over 120 pages total).
Make sure you read a good translation, of which several exist in modern English (as for many people, reading the best translations in old English is difficult, and makes the entire exercise seem frustrating, like the KJV and DRC and to a lesser extent the CCD), like the RSV-2CE (except for the Psalms), the ESV w/ “Apocrypha” (deuterocanonicals), the OSB, the NKJV, the NASB, the Knox, or any of several others. You should see what translation you have, to see if it’s at least usable (most are, even if they’re not good for in-depth Bible study), or if it’s not.
Edit: when I first read the Bible, as a Muslim, I read first New Testament, cover to cover, then Old Testament, cover to cover, and almost didn’t make it through the first three books of the OT (due to Leviticus): I doubt I would have if I wasn’t a reading freak who can, and loves to, read anything, and I wouldn’t recommend others do it in the same way (although it did work for me, as I’m a Catholic now). The entire book (NRSV NT, NJPS OT, probably some of the absolute worst translations on the market - and it still brought me to Christ, as God can even work miracles from nothing at all, if the Holy Spirit opens thy heart), took me about a week. Needless to say, I read it without commentary, and had to go back and re-read it several times to catch more of the prophecies, typologies, and nuances in the infinitely nuanced written Word of God.
The first time I read the Bible it was hard to understand the context, the story, the timeline and the peoples. But having read it many times since it is absolutely the most beautiful thing ever. Last time, the thing that touched me the most was how the sacrifice was set up in the inner temple and how our altar, lampstands and Mass were absolutely Jewish.
The more you read, the more you understand the will of GOD. You can ask your priest about the daily reading of the church during the Mass to be united with the church. I read the daily Bible reading of my church, it is written in the church calender.