Today I learned that the gospels weren't actually written by who they were named after, and they were supposedly not eyewitnesses. Apparently some gospels were based off others, the miracles cannot be confirmed, and there are inconsistencies. If this is the case, how did the gospels become so widely accepted? What parts can be verified as true? Is this the same throughout the entire bible?
This is very controversial territory so I'm going to try and look at this from as balanced a point of view that I can, but be aware, everyone including myself has an agenda.
Only one gospel claims to have been written by an eyewitness: John. Also, I don't know where you've heard this but nobody has been able to verify the actual writers of the gospels, so their is no reason either to trust or distrust their names. John is also the most unique of the gospels, this has led many to speculate that the source of the stories is a different one to the others, and that this source was outside the loop of Christians in the eastern Mediterranean, this would be consistent with the character of John, but like I said, nobody can check.
The others are known as the synoptic gospels, for their synoptic links. The first to be written was Mark, John Mark were the most common hebrew and latin names combined, which only makes it harder to check it's authorship. This gospel was almost entirely composed from one source, legend has it Peter.
The other two, Luke and Matthew were based off two sources, one the same source as Mark and the other is known as Source Q. each of the synoptic gospels contain an additional amount of original material.
Regarding the inconsistencies, you will find thousands of apologist's views on these, the contrasting view is that memories are fragile, and people make stuff up. Really it's down to you to look at these inconsistencies and judge for yourself do they damage the message.
It's impossible to categorically declare something "true" by the witnesses accounts. Miracles are, by nature, miraculous, impossible and unlikely scenarios.
If you are asking, compared to other historical texts how reliable are the new testament writings as historical documents. Unbelievably so, like, there is no other series of events of this time which has 4 accounts written within 50 years and can be cross referenced with other texts including that of opponents. However, this does not once and for all "prove" the events contained in them.
Here is a question I answered about the canonization of the Old Testament: https://www.reddit.com/r/AskHistorians/comments/4hyhxh/how_did_the_canon_of_the_jewish_bible_the_tanakh/
And here's one someone else has written about the canonization of the New Testament: https://www.reddit.com/r/AskHistorians/comments/1xte29/what_factors_went_into_play_when_deciding_which/