What do the letters of Trajan and Pliny tell us about Roman attitudes towards the Christians?

It should be noted that the Christian letters form part of a much larger correspondence between Trajan and Pliny, and this correspondence is primarily about administration and governance. Running throughout the letters is a sense of pragmatism and a sense through which the Romans appear to be highly results-motivated: they're mainly interested in upholding law and order and maintaining proper governance. The letters about the Christians should be understood within this larger context.
The first thing to note is that Christianity was treated as criminal: this is quite clearly stated by both authors. At the same time, there are clear protocols and a sense by which the crackdown on Christians is bounded in Roman legal precedence. It is beyond dispute that the Roman State was hostile to Christianity and was closely aligned with traditional Paganism.
It's clear that the presence of Christianity has created a lot of instability, and this is very worrisome to Roman authorities. Pliny writes about the spread of accusations throughout the province, and sometimes it reads as if this potential instability concerns him even more than the presence of the Christians themselves. At the same time, it should be noted that, when dealing with suspected Christians, he gives the accuser every opportunity to deny the charges, after which the alleged Christian would be released. Then there is Trajan's response to Pliny afterwards, instructing him not to go out of his way in hunting Christians. A complicated picture is beginning to form.
https://sourcebooks.fordham.edu/ancient/pliny-trajan1.asp