Jesus told his followers to go into all the world and "preach" the gospel. He said nothing about writing anything down for a mostly illiterate population with no paper at the time. He said nothing about a future New Testament. He said I will send the Holy Spirit who will teach you.
From the article "Why you can believe the Bible"
The main reason the Gospel accounts were not written immediately after Jesus' death and resurrection is that there was no apparent need for any such writings. Initially the gospel spread by word of mouth in Jerusalem. There was no need to compose a written account of Jesus' life, because those in the Jerusalem region were witnesses of Jesus and well aware of his ministry
http://www.everystudent.com/features/bible.html
ETA And just because we have no contempory writings of Jesus' ministry that doesn't mean they never existed. We have no records of Julius Casesar's signature, that doesn't mean he never signed anything. We have no contempory writings (in existence) of Alexander the Great's life, that doesn't mean he didn't conquer much of the known world. In fact almost all of what we know about his life comes from historians who wrote over 300 years after his death.
Former skeptic and Christian persecutor (the apostle Paul) who approved of Steven's stoning claimed to have saw Christ. Paul wrote Corintinians about 20 years after the Crucifixion. He also wrote at least 12 other New Testament chapters.
Although authorship of the Gospels is debated among scholars, Matthew and John were apostles and thus eyewitnesses, Mark, was an associate of the main apostle Peter, and Luke was an associate of Paul (Paul actually met with the main apostle Peter for about 2 weeks according to the book of Acts). And gospel writer Luke, who has been acclaimed a great historian regarding non-supernatural events, wrote the apostles were witnessing daily in Jerusalem shortly after the Crucifixion even though they were thrown in jail and told not to do it anymore when they got out of jail. So there is historical evidence of the church being continuous from the day of the crucifixion.
This is false, Paul, who wrote almost half of the NT claimed, if Jesus was not raised from the dead, our faith is in vain.
All of the gospels record the resurrection. Also Jesus claims to be pre-existant to Abraham in the gospel, have the power to forgive sins, is the Messiah, and says when you see me you see the father (God). Also Luke records the transfiguration where God is heard by a few apostles saying "This is my beloved Son in whom I am pleased".