Paul's letters were probably first, beginning with 1 Timothy and Galatians. Romans was a middling book, 2 Timothy was probably his last. They range from 52 ad to 62 or 68, depending on what you think about deutero-Pauline scholarship
Of the Gospels, Mark is usually considered to be first, although some have proposed Matthew. Luke is usually dated after Mark and Q, John is usually considered to be last.
The Revelation of John is usually considered to be the latest of the books, and is typically dated to the 80s or 90s.
All of the books of the NT were at one time considered to have some apostolic authorship, although Hebrews is no longer considered to be Pauline.
According the Holman Handbook of the Bible, probable dates for each of the books of the NT is as follows:
I combed through the Holman Handbook of the Bible to pull out a possible chronology. These are more or less in an order, though dates can differ based on when one assumes certain events to have occurred. Some of Paul's letters, for example, seem to indicate he left Rome and travelled westward to Spain in a "later" career - other sources say, no, he was beheaded in 62AD. Revelation could have been penned under Nero (68) or Domition (95). The point is, none of this is an exact science. Still, here would be my summary for when the books of the New Testament were written.
[Author]Book - notes on the time frame
[Jame]James - 'early' James doesn't have much church hierarchy, could be one of the earliest
[Paul]Galatians - probably Paul's 1st epistle, 48-52
[Paul]1 Thessal - 51
[Paul]Romans - 56 or 57AD
[Paul]1 Corinthi- 55AD
[Paul]2 Corinthi- a short time afterwards - 55 to 57 AD, at least one theory says that the later part is actually "3rd Corinthians"
[Petr]Mark - late 50s (Note: Marcan v.s Mattean priority is a big question. I tend to go with Mark being first)
[Paul]Ephesians - could be 57-59 or 60-62, depending on the chronology of Paul's imprisonment
[Paul]Colossians - 60 or 61, about the same time as Philemon
[Paul]Philemon - 60 or 61
[Luke]Luke-Acts - 62AD (near the end of Paul's ministry)
[Paul]Phillippi - 62AD (Paul is in prison)
[Paul]2 Timothy - sounds very much like a farewell letter from Paul
--- Sources differ as to whether Paul was executed in 62, 65, or 68AD ---
[Paul]2 Thessal - (disputed authorship), later in "Paul's" career
[Paul]1 Timothy - 64 to 67 AD
[Petr]1 Peter - early 60s
[Paul]Titus - (disputed authorship) 65 - 68 AD
[????]Hebrews - mid to late 60s
[Petr]2 Peter - mid to late 60s. Shares much in common with Jude
--- The late 60s included a revolt by the Jews, a diaspora, and finally, the destruction of the Temple ---
[Matt]Matthew - early 70s, if you assume Marcan priority. An emphasis on Jewish tradition could be a reaction to the Temple
[Jude]Jude - difficult to date, could be from 65 AD to 80AD
[JOHN]John - sometime between late 60s and 90 AD. John was the youngest of the disciples, and lived to be the oldest
[John]1 John - early 90s (?)
[John]2 John - there is very little to go on, but the early to mid 90s is not out of the question
[John]3 John - there is very little to go on, but the early to mid 90s is not out of the question. Was being quoted by 95 AD.
[John]Revelation- 68AD(possible), 95AD(more likely)