v16 οι ... ενδεκα "the eleven [disciples]" - minus Judas.
το ορος "[to] the mountain" - can mean "into the hills” though we don't really know what mountain. The presence of the article may be particular, possibly alluding to a Moses type mountain where the new Israel meets with their God. Tradition identifies Tabor but symbolism is Matthew's intention not geography.
εταξατο (τασσω) aor. "had told [them] to go" a rather unrecorded conversation between Jesus and his discipels!
v17 ιδοντες (οραω) aor. part. "when they saw [him]" - having seen.
προσεκυνησαν (προσκυνεω) aor. "they worshiped” i.e. they fell down, the word is used to describe an act of adoration, so probably "they did homage to him", Cassirer.
εδιστασαν (δισταζω) aor. "doubted" - hesitated, were uncertain. the word implies a lack of faith, but it is better to understand the response as one of uncertainty.
v18 προσελθων (προσερχομαι) aor. part. "then [Jesus] came" the participle is temporal. It is very rare for Jesus to do the coming, people usually come to him.
λεγων pres. part. "and said" εξουσια (α) "[all] authority" - the word primarily means "freedom of choice", but extends to mean "absolute authority" here. The point of the statement is that Jesus has the authority to commission the mission to the nations.
v19 This verse has prompted endless debate, both in meaning and application. One translation would be; his assertion that he has authority to authorize and enable this mission, to make new disciples from the nations by immersing them in the gospel ("baptizing them in the name") and teaching them to commit themselves in faith. Or he may be reinstating his apostles after their failure to go forth to teach his law and baptize the nations to be a people of God's possession.
πορευθεντες (πορευω) aor. pas. part. "go/having departed”. probably temporal, "while you are going" the commission is to "make disciples” not to "go". The participle is sometimes treated as attendant circumstance, "go and make", but the going is secondary so; "while you are on your way, μαθητευσατε (μαθητευω) aor. imp. act. "make disciples" – or “disciple those you meet”. The two modifying participles explain how disciples are made, but as these notes indicate the how is anything but settled. "As you move through life, introduce everyone you meet, far and near, to this way of life."
βαπτιζοντες (βαπτιζω) "baptizing" - immersing. adverbial, possibly modal, expressing the manner or means by which disciples are made. It is quite misleading to transliterate the Greek wordβαπτιζω into English. Whenever we see the word "baptize" we think of water baptism but the actual meaning is "immerse", either literally in water, or figuratively as in to “overwhelm”, e.g. remember Greek is an idiomatic language, (by that I mean it doesn’t really rain cats and dogs in America) so “immersed” in the Spirit, or “immersed” into suffering seems more than likely especially with the modifying phase "in the name" which means to be placed under the authority of the divine.
So it is likely on Matthew’s community that the water dunking or sprinkling is not the dominant part of the "immersing in the name" – for after all the Baptist came baptizing but Jesus came preaching. The tradition of water baptism is surely set early in the Christian church as a visible expression of repentance, but we need remember Jesus didn't perform baptisms, (Jn.4:2 notwithstanding). In the end, the making of disciples is not achieved by immersing a person in water but by immersing a person into the nature and character of God. This is achieved, not by rites but by words, the communication of the gospel.
εις "in" - το ονομα "in the name". It is unlikely that Jesus is dictating a formula here, but he is offering, al la Moses, a new name for the God who gives Him authority and claims the new community Jesus both envisions and creates. So this theophany is like to that of Sinai and the burning bush –instead of a prophet, God sends the church.
πατροσ και του υοιυ και τουο αγιου πνυεματοσ − father and the son and the holy spirit; the new name for God the new Church will speak to provide authenticity and authority for her message and mission - hence its inclusion on this particular Sunday.
v20 διδασκοντες (διδασκω) pres. part. "teaching" – the second mode expressing how, or by what means, disciples are made. So, "immersing them in the gospel and teaching them."
τηρειν (τηρεω) pres. inf. "to obey" - keep guard.... better than observe or practice – an epexegetic expressing the content of the teaching, namely "that they obey." The word "obey" always leads to "obey the commandments." But the apostles are to instructed to instruct others to obey/keep/guard, the teachings of Jesus, ie., the κεριγμα, the apostolic preaching of the cross.
ενετειλαμην (εντελλομαι) aor. "I have commanded" - in the sense of emphatic or important instructions.
ιδου (οραω) - behold/ take note/ pay attention. εγω pro. "I" – an emphatic by position.
μεθ (μετα) + gen. "with [you]" -expressing association.
πασας τας ημερας "always" – literally all the days i.e. throughout all of recorded history, all of created time.
εως + gen. until. A temporal preposition serving to introduce a temporal construction.
της συντελειας του αιωνος "the very end of the age" - the completion of the age = the completion of God's purposes for the world and humanity.