The answer is in the context, as it usually is. At 1 Corinthians 1:18, Paul begins contrasting the wisdom of the world with the wisdom of God. The statement, "I resolved to know nothing but Christ and Him crucified" was written in that context. Paul resolved to preach the gospel of Jesus Christ, including His crucifixion, which was incomprehensible to the Greek mind that sought wisdom, and a stumbling block to the Jewish mind that sought a powerful and even invincible king who would conquer their worldly enemies.
Paul's statement, in context, means that he wasn't going to try to convince people to enter the Kingdom by means of logic or military might. Rather, he preached the simple gospel: Christ overcame death by dying, which appealed to neither Jews nor Greeks, but is wiser than both.
By the way, when you are trying to understand a verse, it is important that you carefully consider its wording. You stated, "Paul was very adamant that all he needed to know was Christ and Christ crucified." That's not what Paul wrote (unless it's in a translation that I haven't read); Paul's words were, "I resolved to know...."