See the notes at Matthew 8:19-22.

A certain man - He was a scribe. See on Matthew 8:19-22 (note). It is probable that this took place when Christ was at Capernaum, as Matthew represents it, and not on the way to Jerusalem through Samaria.

(13) And it came to pass, that, as they went in the way, a certain [man] said unto him, Lord, I will follow thee whithersoever thou goest.
(13) Those who follow Christ must prepare themselves to endure all discomforts.

And it came to pass, as they went in the way,.... From one village of the Samaritans, to the other; though if this is the same history related in Matthew 8:19 it was as Christ went from Capernaum to the sea side, in order to go to the other side of it; and must be inserted here, without regard to the order of time:
a certain man said unto him; if the same as in Matthew, he is there said to be a "Scribe";
Lord, I will follow thee, wheresoever thou goest. The Arabic and Ethiopic versions read these words by way of question, "Lord, shall I not follow thee wheresoever thou goest?" See Gill on Matthew 8:19.

Here is one that is forward to follow Christ, but seems to have been hasty and rash, and not to have counted the cost. If we mean to follow Christ, we must lay aside the thoughts of great things in the world. Let us not try to join the profession of Christianity, with seeking after worldly advantages. Here is another that seems resolved to follow Christ, but he begs a short delay. To this man Christ first gave the call; he said to him, Follow me. Religion teaches us to be kind and good, to show piety at home, and to requite our parents; but we must not make these an excuse for neglecting our duty to God. Here is another that is willing to follow Christ, but he must have a little time to talk with his friends about it, and to set in order his household affairs, and give directions concerning them. He seemed to have worldly concerns more upon his heart than he ought to have, and he was willing to enter into a temptation leading him from his purpose of following Christ. No one can do any business in a proper manner, if he is attending to other things. Those who begin with the work of God, must resolve to go on, or they will make nothing of it. Looking back, leads to drawing back, and drawing back is to perdition. He only that endures to the end shall be saved.

On these excuses, see notes on Matthew 8:18-22; also Mark 4:35. Luke adds a few new facts which I notice.
A certain man. A scribe (Matthew).
I will follow thee. The Lord tries his sincerity by telling him of the hardships before him.
The Son of man. The Messiah (see note on Matthew 10:23). He had no permanent home. Often his only shelter at night was the open sky.
Another. A disciple. See note on Matthew 8:21.
Let the dead bury their dead. Let the world attend to its own duties. To preach the kingdom rises above every other duty. The disciple was probably an apostle, who wished to stay with his father until his father's death.
No man having put his hand to the plough. An agricultural figure. The plowman must look ahead instead of back. So, too, of the Christian. The Savior's words were suggested by a request of one to go home for a farewell before following him. In thousands of instances it has been shown that the convert who goes to confer with unbelieving relatives before the final step is likely to part with Christ instead. Heathen, ready to be baptized, who go home for a final farewell before the step, are likely to be persuaded not to take it, so missionaries tell us.

Matthew 8:19.