We do not know the exact order in which they were written, and there is vigorous debate in scholarly circles surrounding this topic.
It is generally agreed that most of Paul’s letters are the earliest NT writings. Most date them between 50A.D. and 67A.D. (if you accept they were all written by Paul). One possible dating of his letters are:
1&2 Thessalonians (50-52)
1&2 Corinthians (56-58)
1&2 Timothy (63-67)
The dating of the Gospels is notoriously difficult. Most modern scholars place all but Mark after 70A.D., but that is based on some questionable presuppositions. Personally, I think their order is Matthew, Luke, Mark (all before 70A.D.) and then John, which was perhaps as late as 90A.D. Acts was most likely written in the early 60’s.
The letters and Revelation of John are typically dated between 80A.D.-100A.D.
Hebrews, 1 Peter, and James have suggested dates ranging from the 50s to the 90s. I tend to think each was written before 70A.D.
Jude and 2 Peter are often dated very late - even 2nd century - but it is pretty hard to know when these letters were written.