I recommend Luke/Acts for Jesus's ministry and an overview of the early church, Romans for a more systematic view of salvation, Hebrews to see how Christianity relates to Judaism, and 1+2 Corinthians for a general overview of the Christian life. All of the books are of course important, but those are the good starting points, I think.
For the OT, I recommend the Pentateuch (first 5 books of Moses) for an overview of God's covenant with Israel, God's promises and the foundations of our faith. I suggest a Bible with some good commentary so you get a good feel for why all these odd laws and practices are in place, because they really are important. I recommend Job, Psalms, Proverbs and Ecclesiastes for some amazing wisdom literature regarding life, philosophy and God. Finally, the major/minor prophetic books (Jeremiah, Ezekiel, Amos, etc) for Messianic prophesies and glimpses into how God works with His people.
As for the reading itself, go slowly and carefully; there's a lot to learn and it does require some effort and digging on your part. You have to read like an ancient Hebrew to be able to wrap your mind around the poetic nature of a lot of the writing, so (especially in the OT) look for subtle themes of truth rather than academic discourses. That doesn't come until Paul's epistle to the Romans.
Some excellent resources I use:
A Christian Thinktank - Excellent commentary on pretty much every aspect of the faith.
English Standard Version (ESV) - Biblia.com - Bibles, commentary, and a lot more. Pretty much the only Bible study tool I use online.
Reasonable Faith: - a very scholarly and academic website for apologetics, q+a, philosophy and a lot more.
Main Page - Theopedia, an encyclopedia of Biblical Christianity - basically wikipedia for Christianity.
Above all, don't stop asking questions.
