Are you asking if, in general, Jesus taught that there is life after death? Or are you looking for specific discussion of the word or concept of "spirit"?
If the former: Matthew 22:23,29-32 "The same day the Sadducees, who say there is no resurrection, came to Him and asked Him ... Jesus answered and said to them, “You are mistaken, not knowing the Scriptures nor the power of God. ... But concerning the resurrection of the dead, have you not read what was spoken to you by God, saying, ‘I am the God of Abraham, the God of Isaac, and the God of Jacob’ God is not the God of the dead, but of the living.”" [NKJV]
Jesus clearly said that there is life after death. He also indicated that those who have died are experiencing that eternal life now, not that they will experience it at some point in the future. Of course Abraham and Isaac had physically died many centuries before Jesus said this, so their bodies were dead. So the "them" that is still alive must be some sort of spirit or soul or a new body.
That said, I cannot find anyplace in the Gospels where Jesus talks about a human "spirit" in that sense. Jesus often talks about the Holy Spirit, and about evil spirits. He has a few places where he talks about a person's spirit in the sense of their personality, like Mark 14:38, "The spirit indeed is willing, but the flesh is weak", or John 4:23, "true worshipers will worship the Father in spirit and truth".
But Jesus seems to reject the idea that after death a person becomes a non-corporeal spirit. After his resurrection, when he appears to the discipiles, he says, Luke 24:39, "Behold My hands and My feet, that it is I Myself. Handle Me and see, for a spirit does not have flesh and bones as you see I have."
Well, I suppose you could say that Jesus' resurrected body might not be the same as the bodies that people have after death, I don't know any place where Jesus specifically says that they are. But Paul says that our resurrection will be like Christ's, for example 1 Cor 15:20 "But now Christ is risen from the dead, and has become the firstfruits of those who have fallen asleep." (It might not be clear from that one verse; read the whole context.) If you're asking specifically what JESUS said rather than what the BIBLE says, then I don't know of any place that Jesus specifically describes the resurrection body other than describing his own.
I conclude that Jesus uses the word "spirit" in a more specific sense then we use the word today. I don't think Jesus used "spirit" as a synonym for "soul". He appears to use it to refer to a specific category of being -- the Holy Spirit, angels and demons.