I. Luke 1
The first instance where the Trinity appears is in 1:8-17, in which an angel of the Lord informs Zachariah that his wife will give birth to John the Baptist. The Three Persons of the Godhead are differentiated from One Another as follows:
i. The Lord [v. 15a]: John the Baptist would be great before the Lord, i.e. the Father. We know that this is the Father who is being addressed as the Lord because of the differentiation that is to follow in the Person of the Holy Spirit. The Personhood of the Father is quite obvious.
ii. The Holy Spirit [v.15b]: John’s ministry would be supernaturally guided by the Holy Spirit who would indwell him, empowering him for this task of declaring the coming of Christ the Lord. I think only a crazy person would even suggest that Christ is not a Person.
iii. The Lord [vv.16-17]: John would also turn the hearts of the people to the Lord their God. Now, in case one says that this is a reference to the Father, the angel goes on to state that John will “go before Him in the spirit and power of Elijah.” And this is exactly what our Lord declared of John’s ministry when He told the disciples:
“…Elijah has already come, but they did not recognize him, but did to him whatever they pleased. So also the Son of Man will certainly suffer at their hands.”
The Personhood of the Holy Spirit is evident from the following facts: (a.)He is differentiated from the Father and the Son, (ii.)He acts as indwelling God, leading John the Baptist, and (c.) He is differentiated from “the power of the Most High.”
Thus, we see the Father as He whom John walks before, the Holy Spirit as He who empowers John for the task of declaring the Son, and the Son who is the Savior. This is the Trinity very plainly, if one has eyes to see the truth of God.
The very beginning of Luke’s Gospel, therefore, opens with a reference to the Triune God, Father, Son, and Holy Spirit. As we move on to the Good News given to Mary by the angel Gabriel, we again see the Three Persons of the Godhead revealed even more clearly in 1:26-35. In this short passage of Scripture, we encounter the Father, the Son, and the Holy Spirit working in unison, as the Father unconditionally elects Mary to be overshadowed by the Holy Spirit for the express purpose of bearing the Son of God. The Father, strictly speaking, is electing/decreeing, the Son is entering into our humanity in order to suffer in our place, and the Spirit is bringing these things to fruition.
The third instance of the Trinity in Luke 1 appears in vv.39-45. Elizabeth’s words, in particular those words given in vv.41-45, show us that the Holy Spirit causes Elizabeth to speak the truth about Christ, Christ is her Lord (v.43), and the Father is also her Lord who has decreed these things, promising them and fulfilling His promises (v.45). Again, the Triunity of God is especially clear.
Finally, the fourth instance in which we observe the Trinity is in 1:67-75, where we learn that (i.)the Holy Spirit causes Zacharias to prophesy and thank (ii.)God the Father for fulfilling His promise in sending (iii.)Messiah, the Son of David, to save men from their sins. Again, the consistency is overwhelming. The Father is addressed as the Lord who has elected/decreed these things to come to pass; the Son is identified as the Agent of Salvation; and the Holy Spirit is the Spirit of Prophecy/Spirit of Truth who causes Zacharias to speak the Truth.
II. Luke 2
The doctrine of the Trinity appears twice in Luke 2:26-28. We can see this in that: (i.)the Father is referred to as “the Lord,” (ii.)the Son is referred to as “Christ,” and (iii.)it is the Holy Spirit who reveals Christ prophetically and in the present. In the second instance, we learn that (i.)God the Father is blessed for fulfilling His Word in (ii.)Christ, and that (iii.)such knowledge is given to Simeon by the Holy Spirit. All Three Persons are mentioned and described according to their external acts of (i.)Election/Decree, (ii.)Substitutionary Sacrifice/Salvation/Redemption, and (iii.)Revelation/Guidance/Sanctification. Regarding the Personhood of the Holy Spirit, it is impossible for a force or current of Divine energy to reveal anything, since the Biblical doctrine of revelation is always Personal, as it comes from the Lord of Hosts; therefore, the Holy Spirit’s revelation is a sure indication of His Personhood, and stands alongside the Father’s election/decree and the Son’s incarnation and salvific work on the cross of Calvary.
III. Luke 3
This is, perhaps, one of the more obvious passages where we see Father, Son, and Holy Spirit, as (i.)Christ is baptized, (ii.)the Holy Spirit descends upon Him, and (iii.)the Father proclaims that Christ is His Son in whom He is well pleased. The ad extra acts of the Three Divine and Holy Persons of the Godhead are clearly seen here as well, where (i.)the Father declares Christ to be His Son/Messiah/Promised Deliverer/Promised Seed, (ii.)the Son is baptized in order to fulfill all righteousness as our Substitutionary Sacrifice, and (iii.)the Holy Spirit rests upon Christ, leading Him, communing with Him, etc.
Christ’s time in the wilderness is specifically related to His being tempted, as a Man, to sin against the Father. Therefore, any reference to God in this context is primarily of the Father, secondarily of the Spirit, and via implication also about the Son, as the source and giver of absolute moral Law. Thus, we learn that (i.)Jesus is tempted to sin against (ii.)God the Father, although He has been filled with (iii.)the Holy Spirit who has driven Him to the wilderness in order to be tempted and fulfilled prophecy by not sinning. This is the first instance of the Trinity in Luke 4.
In the second instance, when the Lord Jesus opens the scroll of Isaiah, He reads from an explicitly Trinitarian passage, in which (i.)the Father has anointed (ii.)the Son with (iii.)the Holy Spirit. Note, once again, the consistency of the actions attributed to each of the Divine Persons. Namely: (i.)the Father decrees/elects/anoints, (ii.)the Son preaches, heals, sets the captives free, and (iii.)the Holy Spirit rested upon Christ without measure.
V. Luke 10:21
As far as I can tell, this is the next instance in the Gospel of Luke where the Trinity appears. In particular, verse 21 explains that: (i.)the Father decrees/elects, (ii.)the Son reveals God to us, and (iii.)the Holy Spirit is rejoiced in.
VI. Luke 11:13 & 20
The next two instances are particularly short, occurring in verse 13 and teaching us that (i.)the Father, according to (ii.)Christ, will give (iii)the Holy Spirit to those who ask Him; and also occurring in verse 20, where we read that (i.)Christ casts out demons by the power of (ii.)the Finger of God (which we learn from cross referencing this passage to Matthew 12:28 is the Holy Spirit), and brings the kingdom of (iii.)God upon the earth.
VII. Luke 12:8-12
We find the first reference to the Holy Trinity in 12:8-12, in which the (a.)Christ proclaims that blasphemy against Him is pardonable, but blasphemy against (ii.)the Holy Spirit is unpardonable, and informs us that He, Christ, will stand before (iii.)the Father, who will acquit or condemn individuals on the basis of Christ’s work.
VIII. Luke 20:41-44
Here we need to cross reference Mark 12:35-37 in order to see the whole Godhead united here, as (i.)the Father appoints the Son to sit at His right hand as heir of all things, (ii.)Christ is declared to be the Son of God with power, seated at the right hand of the Majesty on high and crushing His enemies underfoot until the time of consummation occurs, and (iii.)the Holy Spirit is He who spoke by the mouths/through the mouths of the prophets/Scripture, and who pointed to Christ’s Deity, His equality with the Father and His incarnation as the Substitute for God’s elect people.
IX. Luke 24:29
Here we encounter the last presentation of the Trinity. (i.)Christ declares that He is risen and has fulfilled all prophecies concerning Himself and salvation, etc, (ii.)the Holy Spirit will come to fill them with power from on high, He is being sent by (iii.)the Father who has called the Holy Spirit “the Promise,” i.e. the Promised other Comforter.
 Cf. Matt 17:12
 As an aside, one sees the futility of Satan and his minions in attempting to overthrow the Gospel. The Father has elected the Son and has given Him the Spirit without measure.
 I think it may be debatable as to whether or not the whole Godhead is signified here, or if the Lord Jesus is simply speaking about the Father. On the basis of the analogy of Scripture, I believe that our Lord is speaking about the Father, in particular, but that this passage also has greater significance as it applies to the Father in election, the Son in redemption, and the Spirit in sanctification.
 Cf. John 14:15-18