Some scholars have a problem believing Paul ever intended to visit Jerusalem when he left Corinth, because neither he nor Luke mentions that intention. I love to study scholarly reviews of the text, and see the things that they see, because my eyes are simply not trained to pick these things out. But, I almost pity them in their search of exactitudes that will permit them to believe this or that about the text.
If I bake a birthday cake, complete with candles and the writing on top, the natural implication would be that either I baked it for a friend who intends to celebrate someone’s birthday, or I intend to use it to celebrate someone’s birthday myself, even if I never mention the intention forthrightly. The fact a birthday cake was baked infers that a birthday celebration will take place somewhere by someone or a group, and this birthday cake will be used in that celebration. Is this not a logical conclusion—even if I never mention that I, or someone for whom I baked the cake, intend to celebrate a birthday? I am simple enough to believe this would be true.
Luke mentions in Acts 18:18 that after a long while Paul took leave of the brethren at Corinth and sailed to Syria. From this some scholars conclude Paul intended to sail to Antioch and conclude his 2nd recorded missionary journey there. In fact, some conclude that the only reason Paul set out from Ephesus to Caesarea instead of Antioch was that either the winds weren’t good for sailing to Antioch or Paul couldn’t book a vessel for Antioch at the desired time (cp. Acts 18:21-22).
Let’s back up to verse-18. Luke also tells us that Paul shaved his head because he had a vow. This mention of Paul shaving his head makes no sense at all, unless he intended to take his locks to Jerusalem within 30 days and offer them with the appropriate sacrifices in the Temple there. Why else would a Jew shave his head to conclude his vow? What else would he do with the shaved locks? Is there another way to connect the dots?
Secondly, just because Luke mentions that Paul sailed off for Syria from Corinth (Acts 18:18) does not mean he must have intended to go directly to Antioch from Corinth. How do we know this? Well, Jerusalem is included in the Roman province of Syria. Paul sailed from the Roman province of Achaia intending to arrive at the Roman province of Syria. Matthew tells us that Jesus’ fame went throughout Syria, and people brought him all their sick (Matthew 4:24). If we are to understand this verse like some scholars read Acts 18:18, then Jesus must have been healing in Antioch. Does this make sense? It doesn’t make any sense at all to me, and it shouldn’t to anyone else who reads the Scriptures with a desire to know what they actually say.
Upon his arrival at the seaport in Caesarea Paul went up (G305) to greet the church—and then went down (G2597) to Antioch. What does this mean? Admittedly, Jerusalem isn’t mentioned, but how should one connect the dots? Whenever a city is mentioned in the Scriptures using this Greek word (G305 anabaino, ‘ascend, arise, go up etc.’), it is always Jerusalem the writer has in view (Luke 2:4, 42; 18:10; John 2:13; 5:1; 7:10, 14; 11:55; Acts 21:15; 24:11). Similarly, whenever a city is mentioned with the Greek word meaning to come down (G2597 katabaino), the word is always connected with traveling from Jerusalem (cp. Mark 3:22; Luke 2:51; John 4:47; Acts 23:10; 24:1, 22; 25:6). Is this too simplistic a view for one trained to derive truth from the text? I hope not, because, quite frankly, it is the only image I am able to see emerging from what I read at this point in Acts 18.
 See Robert H. Smith; Concordia Commentary: Acts.