How close to the original texts is the New Testament today?

I guess I'd tackle this question by first addressing what "in the texts today" means. Biblical versions in modern languages are translated from what's called critical editions of the NT. A critical edition is produced by taking the earliest manuscripts of the NT we have, comparing their texts and--through certain (quite complicated and contentious) processes--determining the common "archetype" for these manuscripts, which is thought to be close(r) to the original texts. (Big caveat on the term "original text" here, if you're looking to get waaay into some hardcore stuff.).

Much more could be said about this process, but the salient point is that the "texts today" are really based on manuscripts (or reconstructed texts) from the 2nd-5th centuries--which doesn't lie too far away from the time of their original composition. (Though there are many other texts--for example the Homeric works, etc.--for which the earliest manuscripts are ~1,500+ years after the time of their original composition, and yet are almost certainly very close to the "original" texts.)

The question is then this: if we can reconstruct, with good confidence, what the NT texts looked like in, say, the (mid-to-late) 2nd century, how do we know that the texts weren't changed in the time between their original composition (in the mid-1st to early 2nd century) and then?

There are some general methods for determining this. One of the main ways in that interpolations into the texts are often times disruptive to their style/vocabulary/syntax/flow, etc. Perhaps if one were unscrupulous enough to actually attempt to alter these texts, they were at the same time careless enough to not spend a lot of time making sure it wasn't an obvious intrusion into these elements of the text.

But there are other cases in which there is no consensus as to whether something is a secondary interpolation or not. Some of the most hotly-debated passages here include things like 1 Corinthians 14:34-35 and 2 Corinthians 6:14-7:1. People can be trigger-happy about proposed interpolations, though; and you can find discussion/analysis of tons of proposed interpolations in books like William Walker's Interpolations in the Pauline Letters.

Finally: there are a few instances in the Hebrew Bible where what we think of today as a single "book" was originally a shorter original work, with later redactors then taking another (later) related work and adding it to the original one, so that it's almost like a "collection" of books within a single one. For example, in the Wiki article for the book of Isaiah, you can find this:

The scholarly consensus which held sway through most of the 20th century saw three separate collections of oracles: Proto-Isaiah (chapters 1–39), containing the words of Isaiah; Deutero-Isaiah (chapters 40–55), the work of an anonymous 6th-century author writing during the Exile; and Trito-Isaiah (chapters 56–66), composed after the return from exile. While one part of the consensus still holds – virtually no one maintains that the entire book, or even most of it, was written by one person – this perception of Isaiah as made up of three rather distinct sections underwent a radical challenge in the last quarter of the 20th century.

(The book of Enoch is another great example of this, with the divisions between the separate works that comprise the "one" book even more obvious.)

However, proposals of a similar process taking place in the NT have not held large sway. Yes, it's virtually uncontested that, say, the author of the gospel of Matthew copied large sections of the gospel of Mark; but this is merely a utilization of another source--it's not like Matthew is just Mark + another autonomous work tacked onto it.

Perhaps the most comparable thing in the NT is in the composite nature of things like the Second Epistle to the Corinthians: that this "single" epistle was actually several separate letters that were then compiled into one "collection."