25 ἐξῆλθεν δὲ εἰς Ταρσὸν ἀναζητῆσαι Σαῦλον,
Now he went to Tarsus to look for Saul,
δὲ or “Now” is a conjunction that usually indicates a contrast or simple coordination with what precedes the verse. In this case, it seems that the intent is not a contrast but is stronger than a simple connective. Perhaps an inferential conjunction showing a conclusion or deduction from the previous section better captures the sense. The intended idea, then, would be “therefore” or “so.” This verse and the following content is a result of that which naturally follows for Barnabas based on what he witnessed in the church and the need they had.
ἀναζητῆσαι Σαῦλον or “to look for Saul.” The infinitive ἀναζητῆσαι indicates purpose. The phrase shows the reason that Barnabas went to Tarsus. Some scholars have indicated that the verb used indicates that Barnabas had some trouble finding Saul. He had to “search for him” as it were.  Regardless, Barnabas knew where Saul was because he had part in helping him escape Jerusalem and sending him to Tarsus (9:30), and he went for the express purpose of “finding him.”
26 καὶ εὑρὼν ἤγαγεν εἰς Ἀντιόχειαν. ἐγένετο δὲ αὐτοῖς καὶ ἐνιαυτὸν ὅλον συναχθῆναι ἐν τῇ ἐκκλησίᾳ καὶ διδάξαι ὄχλον ἱκανόν, χρηματίσαι τε πρώτως ἐν Ἀντιοχείᾳ τοὺς μαθητὰς Χριστιανούς.
and after finding [him], he brought [him] to Antioch. And it was for them even for a whole year that [they] were gathered together with the church and taught a considerable crowd, so in Antioch the disciples were first called Christians.
καὶ εὑρὼν or “and after finding [him]” is probably a temporal aorist participle indicating antecedent time. In other words, “having found” or “after finding” him, Barnabas then brought Saul back with him to Antioch after completing the purpose or goal for which he went in the first place, that is to gain needed help to meet the identified need of the church in Antioch.
δὲ or “and” like at the beginning of verse 25 is a conjunction that usually indicates a contrast or simple coordination with what precedes the verse. Unlike in verse 25, taking the conjunction as a simple connective, which simply indicates an addition to the previous discussion, is the best option for its use in verse 26. The context here suggests that the next clause of verse 26, “it was for them for a whole year, that [they] both were gathered together with the church and taught a considerable crowd,” is the natural progress, addition, or next step to Barnabas finding Saul and bringing him to Antioch.
ἐγένετο or “it became” is the only indicative verb in the Greek sentence. All of the other verbal forms are infinitives. Some scholars propose that ἐγένετο serves as the main verb of the sentence and the following infinitives are functioning as a compound subject:  i.e. “being gathered together with the church” and “teaching a considerable crowd” became. For smoother reading, the verb has been translated as “was” here with an understood “it” as the subject. The two following infinitives are presented as content clauses: “that [they] both were gathered together with the church and taught a considerable crowd.” Whether the infinitives are taken as subjects or content clauses does not functionally change the meaning. The bigger question is the significance of the actions indicated in the two infinitive phrases.
αὐτοῖς or “for them” is a dative of advantage. The referent of the personal pronoun “them” most likely is the church. Therefore, the idea is that the activities that Barnabas and Saul engaged in, i.e. that which is indicated by the verbal infinitives, is for or to the advantage of the church.
ἐνιαυτὸν ὅλον or “a whole year” refers to a specific length of time. It was for a whole year, for the benefit or advantage of the church, that these identified activities were engaged in.
συναχθῆναι or “to have been gathered together” is a passive infinitive best translated “were gathered.” ἐν τῇ ἐκκλησίᾳ or “among” or “with the church” indicates with whom it was that Barnabas and Saul gathered together. This is significant for at least two reasons. First, the focus of this activity was not to the lost in the city but the redeemed. Second, as will be indicated by another portion of the passage below, the ministry seems to have occurred in a public large group setting rather than with groups or individuals. Therefore, for the length of an entire year, Barnabas and Saul were gathered together with the church in Antioch.
διδάξαι ὄχλον ἱκανόν or “to have taught” is an active infinitive translated here “taught.” ὄχλον ἱκανόν or “a considerable crowd” gives more credence to the position that Barnabas’ and Saul’s activity occurred in a public setting rather than with small groups or individuals. For the length of an entire year, Barnabas and Saul publicly gathered with and taught the church in Antioch. Perhaps the better question is what was the content of their teaching. The content is at least implied by the term διδάξαι. Earlier in Acts after Pentecost, Luke offers a statement summarizing the church’s activities in Jerusalem (2:41-47). In verse 42, Luke describes one of the functions as devoting themselves to “the apostles’ teaching.” διδαχῇ in Acts 2:42 is the noun that relates to the infinitive in Acts 11:26. Furthermore, one of the means that Jesus gave in Matthew 28 for accomplishing the Great Commission is “teaching them to observe all that I have commanded you.” The “Apostles Teaching” described in 2:42 then, is most likely what Christ had instructed his followers to teach others to obey, which is the same that Saul and Barnabas taught here. The content of their public teaching must have been what Christ had taught His disciples and instructed them to teach others. By implication, Saul and Barnabas taught the gathered church the Scriptures for an entire year.
χρηματίσαι τε πρώτως ἐν Ἀντιοχείᾳ τοὺς μαθητὰς Χριστιανούς or “so in Antioch the disciples were first called Christians” is an adverbial infinitive phrase indicating result. The result of Saul and Barnabas’s yearlong public teaching ministry in Antioch was that the disciples were called Christians there. In this case, τοὺς μαθητὰς or “the disciples” is a noun in the accusative case functioning as the subject of the infinitive. χρηματίσαι or “were called” is an infinitive and functions as the main verb in this clause. Since it is in the active voice, or ingressive as one scholar has noted, “came to bear the name” may be a more literal translation. For readability and smoothness of translation, “were called” has been given here. As mentioned in the historical analysis of this passage, the designation “Christian” only occurs twice in the New Testament other than the passage under consideration – Acts 26: 28 and 1 Peter 4:16. In all three locations, scholars believe the occurrence was not a self-identification. In other words, outsiders called the followers of Christ “Christians.” “In all three instances, it is a term used by outsiders to designate Christians. Evidently, the term was not used originally by Christians referring to themselves.” As a result, apparently the term signaled something observable concerning those who were given the label. Luke adds to the significance of this event by the use of the term πρώτως or “first.” They were “first called Christians” here. It should be no surprise that being identification with Christ resulted in the church that followed the instruction of Christ.
 Polhill, 273.
 R. C. H. Lenski, The Interpretation of The Acts of the Apostles (Minneapolis: Augsburg, 1961), 457.
 Polhill, 273.