Did Jesus teach the Trinity?

YES

Though Jesus did not explicitly said that the Father, the Son and the Holy Spirit are one, Jesus clearly said that God is his Father and that he is the Son of God and that The Father and The Son are One.

I and the Father are one. (John 10:30, NIV)

Jesus answered: "Don't you know me, Philip, even after I have been among you such a long time? Anyone who has seen me has seen the Father. How can you say, 'Show us the Father'? (John 14:9, NIV)

The words of Jesus also suggests that there is a strong relationship between the Holy Spirit and The Son.

But when he, the Spirit of truth, comes, he will guide you into all the truth. He will not speak on his own; he will speak only what he hears, and he will tell you what is yet to come. He will glorify me because it is from me that he will receive what he will make known to you. All that belongs to the Father is mine. That is why I said the Spirit will receive from me what he will make known to you.” (John 16:13-15, NIV)

Though Jesus did not explicitly said that the Holy Spirit and the Son are one, it is clear enough that the Holy Spirit is doing only the will of the Son, which is the will of the Father. Therefore,

the Father sent the Son, and

the Son sent the Holy Spirit

It depends on who you ask. That's because it depends upon how you interpret the meaning of certain Biblical passages.

Both Trinitarians and Non-trinitarians feel very strongly that their doctrine is taught clearly in the Bible. Trinitarians will answer, "Yes, Jesus did teach the doctrine of the trinity." Non-trinitarians will answer, "No, Jesus never taught the doctrine of the trinity." Both will provide biblical support for their arguments. So, your question is not really a matter of whether or not Jesus did or did not teach it. The question really is, whose interpretation, of those Biblical passages in question, do you find to be the most convincing.

Below are some of the most common examples that Trinitarians will point to, where they claim that Jesus teaches their doctrine. Non-trinitarians each have slightly different interpretations of these passages depending upon their particular doctrines (non-trinitarian doctrines can range anywhere from Sabellianism to Adoptionism, so it's difficult to give a single definitive non-trinitarian interpretation of these passages).

Matthew 28:18-20 (NASB)

And Jesus came up and spoke to them, saying, “All authority has been given to Me in heaven and on earth. Go therefore and make disciples of all the nations, baptizing them in the name of the Father and the Son and the Holy Spirit, teaching them to observe all that I commanded you; and lo, I am with you always, even to the end of the age.”

John 5:17-18 (NASB)

But He answered them, “My Father is working until now, and I Myself am working.” For this reason therefore the Jews were seeking all the more to kill Him, because He not only was breaking the Sabbath, but also was calling God His own Father, making Himself equal with God.

John 8:48-59 (NASB)

The Jews answered and said to Him, “Do we not say rightly that You are a Samaritan and have a demon?” Jesus answered, “I do not have a demon; but I honor My Father, and you dishonor Me. But I do not seek My glory; there is One who seeks and judges. Truly, truly, I say to you, if anyone keeps My word he will never see death.” The Jews said to Him, “Now we know that You have a demon. Abraham died, and the prophets also; and You say, ‘If anyone keeps My word, he will never taste of death.’ Surely You are not greater than our father Abraham, who died? The prophets died too; whom do You make Yourself out to be?” Jesus answered, “If I glorify Myself, My glory is nothing; it is My Father who glorifies Me, of whom you say, ‘He is our God’; and you have not come to know Him, but I know Him; and if I say that I do not know Him, I will be a liar like you, but I do know Him and keep His word. Your father Abraham rejoiced to see My day, and he saw it and was glad.” So the Jews said to Him, “You are not yet fifty years old, and have You seen Abraham?” Jesus said to them, “Truly, truly, I say to you, before Abraham was born, I am.” Therefore they picked up stones to throw at Him, but Jesus hid Himself and went out of the temple.

John 10:25-38 (NASB)

Jesus answered them, “I told you, and you do not believe; the works that I do in My Father’s name, these testify of Me. But you do not believe because you are not of My sheep. My sheep hear My voice, and I know them, and they follow Me; and I give eternal life to them, and they will never perish; and no one will snatch them out of My hand. My Father, who has given them to Me, is greater than all; and no one is able to snatch them out of the Father’s hand. I and the Father are one.” The Jews picked up stones again to stone Him. Jesus answered them, “I showed you many good works from the Father; for which of them are you stoning Me?” The Jews answered Him, “For a good work we do not stone You, but for blasphemy; and because You, being a man, make Yourself out to be God.” Jesus answered them, “Has it not been written in your Law, ‘I said, you are gods’? If he called them gods, to whom the word of God came (and the Scripture cannot be broken), do you say of Him, whom the Father sanctified and sent into the world, ‘You are blaspheming,’ because I said, ‘I am the Son of God’? If I do not do the works of My Father, do not believe Me; but if I do them, though you do not believe Me, believe the works, so that you may know and understand that the Father is in Me, and I in the Father.”

John 14:9-17 (NASB)

Jesus said to him, “Have I been so long with you, and yet you have not come to know Me, Philip? He who has seen Me has seen the Father; how can you say, ‘Show us the Father’? Do you not believe that I am in the Father, and the Father is in Me? The words that I say to you I do not speak on My own initiative, but the Father abiding in Me does His works. Believe Me that I am in the Father and the Father is in Me; otherwise believe because of the works themselves. Truly, truly, I say to you, he who believes in Me, the works that I do, he will do also; and greater works than these he will do; because I go to the Father. Whatever you ask in My name, that will I do, so that the Father may be glorified in the Son. If you ask Me anything in My name, I will do it. “If you love Me, you will keep My commandments. I will ask the Father, and He will give you another Helper, that He may be with you forever; that is the Spirit of truth, whom the world cannot receive, because it does not see Him or know Him, but you know Him because He abides with you and will be in you.

John 16:12-15 (NASB)

I have many more things to say to you, but you cannot bear them now. But when He, the Spirit of truth, comes, He will guide you into all the truth; for He will not speak on His own initiative, but whatever He hears, He will speak; and He will disclose to you what is to come. He will glorify Me, for He will take of Mine and will disclose it to you. All things that the Father has are Mine; therefore I said that He takes of Mine and will disclose it to you.

Those, and other passages like them, would be the closest Jesus came to directly teaching the doctrine of the Trinity. It is impossible to fully support the entire orthodox doctrine of the Trinity using only the teachings of Jesus. Supporting the entire doctrine on a Biblical-basis requires incorporating many other passages from both the Old and New Testaments. There is no place in the Bible where all of the aspects of the doctrine are expounded in a single passage. The word "trinity" is certainly never used in the Bible.

As far as whether or not the passages above constitute what you refer to as "giving a lesson regarding the subject matter," I will leave that judgment call up to you.