Hey jruiz,
Important thing to keep in mind as you go through this - doubt is normal. Do not fear doubt, but use it as an opportunity to learn and to ponder.
Thankfully, there are a lot of really smart Christian people who have thought a lot about these sorts of things. I myself have spent a lot of time (and some money!) on exploring the field of textual criticism, and trying to get to the bottom of the Bible, how it came together, and whether it can be trusted. I am happy to answer specific questions on that front, but I fear if I try and tackle everything at once, it'll be super long and unreadable and unhelpful
One small comment on the Qur'an, though - the problem with Islamic textual criticism is that it doesn't exist in the same way as biblical textual criticism, there are precious few Islamic textual critics, and many Muslims believe that a critical view of the construction of the Qur'an is not required and should not be pursued on purely theological grounds.
A key difference in biblical and quranic textual studies is the actual histories of the text. Early on in the history of Islam, a number of divergent texts of the Qur'an were destroyed by a caliph named Uthmann for theological reasons (in particular, the need to have only one 'perfect' text), in what is know generally as the Uthmannic recension. This did not happen to the Bible - we have a wide variety of texts, and there was never any concerted Christian attempt to destroy divergent manuscripts. This puts the two studies on different footings - we can never be certain of what the original Quranic text was without trusting that Uthman got it right. We can get closer in biblical studies, because we have earlier and more widespread textual witnesses.
In general, though, we can actually see where things were changed in biblical manuscripts, because they still exist and we can study them. The changes are, by an overwhelming majority, differences in spelling or clear transcription errors, many of which cannot even be translated into English. Those that are meaningful can be counted on your fingers, and almost all of those can readily be identified as genuine or non-genuine. The problem is not as big as Muslims often make out, and their overemphasis is typically based on their belief that the true word of God must have always been preserved perfectly (word for word), which is not the Christian understanding. The purity of the Qur'an, in any case, is somewhat artificial because of recessional activity.
I don't know what reading level you would consider yourself to be at, but a fantastic book to read would be Dr Richard Bauckam's '
Jesus & the Eyewitnesses', which deals with some of the issues you've raised. It's at a fairly scholarly level, but not overly technical. Very dense, though, so if you do have a go at it, prepare to be in for the long haul