Luke to Acts
Jesus leaves the planet.
Ascension
The first connection between Luke-Acts is the ascension of Jesus. Luke ends with this event and the first verse of Acts refers to it.
Wait for the Spirit
One step away in the symmetry is Jesus telling his disciples to wait in Jerusalem for the holy spirit.
Witnesses to Jesus
One more step away in the symmetry Jesus tells his disciples they will witness to him in the whole world, beginning in Jerusalem.
Authorship
The authorship of Luke-Acts appears to be possible to determine from scripture. That they are written by the same author(s) is clear from the way both are addressed to one called Theophilus.
The identity of Theophilus is unclear as he's never mentioned anywhere else in the Bible. It's also unique for a Gospel, like Luke, or a book with a large scope, like Acts, to be addressed to an individual. Normally shorter letters are addressed to a group or an individual and large, sweeping narratives, are not addressed to anyone in particular.
One theory for why Luke-Acts is addressed to an individual is that this two volume work is Paul's court case for his hearing in Rome. I like this theory because, if right, it not only explains that the recipient is someone in Rome, but it makes Paul the author, which explains a lot of things.
Not an Eyewitness
First, the author of Luke-Acts was not an eyewitness of Jesus, but instead, had to investigate the stories and interview the eyewitnesses before writing his/her account. Since Paul was not an eyewitness of Jesus' life or ministry, this makes sense.
Scope
After laying the foundation of Jesus' life and ministry, the author takes the story forward to the point when Paul arrives in Rome. If Paul is the author and he's explaining his story to someone in Rome, perhaps as part of his trial, it makes sense that he would end the story at his arrival in Rome.
Caesar's Saints
It's traditionally believed that Paul wrote the Philippians from Rome, but this idea is supported by the way Philippians follows Acts in the book order. Paul's period under house arrest in Rome, told us at the end of Acts, is the backdrop for reading Philippians and makes sense of various details in the letter like mention of the saints in Caesar's house.
We know Theophilus is a convert, or saint, because of the address at the beginning of Luke, and we know Paul was in contact with believers in Caesar's house from Philippians. It's likely, if not provable, that Theophilus is one of those "in Caesar's house." This in and of itself does not prove Paul wrote Luke, but it increases the odds that he is the author.
This argument is perhaps stronger for me than most because of the proximity of Philippians to Luke-Acts in the book order. I'm learning to trust nearby books to contain the clues needed to unpack the book in question. Context across books if you will.
My Gospel
Another scriptural clue that Paul is likely the author of Luke-Acts comes from several references he makes to "my gospel." Here's one.
To summarize, Paul writes Timothy and says that according to "my gospel" Jesus rose from the dead and was a descendant of David.
The first realization here is that Paul is probably actually referring to a written account when he says "my gospel" and not just his way of talking to people about Jesus. He means he authored one of those four books that tell the gospel.
Of the four Gospels Luke is the best candidate to have been authored by Paul because he was not an eyewitness, because it comes with Acts, which is primarily Paul's story, and because of this statement he makes to his disciple Timothy. He says my gospel is special because of the way it deals with the descent of Jesus from David. Does Luke do that uniquely? Yes.
On the strength of the words of the angel Gabriel Jesus is descended from David. This is unique to Luke. Luke also has a genealogy, but Matthew does as well, so that is not unique, but does increase the odds again.
Orderly Account
The author of Luke says to his recipient Theophilus that he's written an orderly account. What does that mean? Whatever all the various possible meanings, one explanation that makes lots of sense is the way Luke-Acts continually hangs the narrative on the times and reigns of different caesars or government officials. For example...
and...
and...
None of the other Gospels refer to Caesar or other governmental people to frame their narrative. This makes sense, though, in Luke-Acts if Paul's providing a written account of his story for Caesar.
Peter's Encounter with Jesus
Paul tells the Corinthian believers that the Gospel he preaches and they are to keep in remembrance says Peter was the first to see Jesus after the resurrection (this is excepting the women who saw him first).
The only Gospel that records Peter seeing Jesus before the 11 is Luke.
The context to this passage in Luke is that two men on the road to Emmaus had encountered Jesus after the resurrection and returned to Jerusalem to tell the 11, but upon their return found that Peter (Simon) had seen him as well. No details are given about Peter's encounter and it's unclear if Peter saw Jesus before or after the two, but it's likely he saw Jesus first given his role at the time. In any case, Luke is the only Gospel that tells us Peter saw Jesus before the others. The question is why is Paul so supportive of Luke as opposed to the other Gospels, but if he wrote it that way after much interviewing and investigating, then obviously he would preach it that way.
Timothy
One more angle on this theory of authorship should include the likely role of Timothy as a co-author. There are a handful of clues that at least make this plausible.
First, there's the "we" pattern in Acts that begins after Timothy joins Paul's travels. The use of "we" does not prove Timothy, though, because others also traveled with Paul, but whoever wrote Acts and referred to "we" appears to have made it all the way to Rome with Paul, and it appears only one or two did so when the Acts narrative is read carefully.
That Timothy was in Rome with Paul is easy to establish to the extent that it's possible to establish that Philippians was written from Rome, because Timothy is named as a co-author of Philippians.
Timothy also co-authored a number of other letters with Paul, which helps establish his fitfulness to the holy spirit for penning scripture.
In terms of a resume he also has this commendation from Paul about his knowledge of scripture.
More than just being a co-author of a New Testament letter, his knowledge of the Hebrew scriptures would have been particularly valuable in co-authoring Luke and Acts, which are fulfillments of prophecy and often quote and explain passages from the Hebrew scriptures.
And it's the case that Timothy was familiar with Paul's Gospel, for the only time Paul mentions it outside of his correspondence with Rome, is while writing Timothy. As quoted before.
If Paul wrote Luke-Acts for court, and Timothy was with Paul in Rome as a witness, then it stands to reason he helped Paul with the written account as well.
All the clues about authorship given above do not prove that Paul and Timothy wrote Luke-Acts, but they do support this theory. In contrast, the usual answer that Luke wrote Luke is not supported with scripture at all, but only by the shared name. This same mistake is made when identifying the author of the Gospel of John, but in that case it's easy to establish from scripture that the author was not the disciple John, but Lazarus. So the name of the Gospel, does not always share the name of it's author.