Jesus taught that we should love all people, and that includes those closest to us
Luke 14:26 (RSV) If any one comes to me and does not hate his own father and mother and wife and children and brothers and sisters, yes, and even his own life, he cannot be my disciple.
The same Gospel writer records Jesus healing Peter’s mother-in-law:
Luke 4:38-39 And he arose and left the synagogue, and entered Simon’s house. Now Simon’s mother-in-law was ill with a high fever, and they besought him for her.  And he stood over her and rebuked the fever, and it left her; and immediately she rose and served them.
Obviously, then, Jesus was expressing himself in a non-literal way in Luke 14:26. Ancient Hebrew and Greek contained a richness of literary forms and genres that every language has.
Bible scholar E. W. Bullinger catalogued “over 200 distinct figures [in the Bible], several of them with from 30 to 40 varieties.” That is a statement from the Introduction to his 1104-page tome, Figures of Speech Used in the Bible (London: 1898). It can be located online for free.
Bullinger devotes six pages to “Hyperbole; or, Exaggeration”: which he defines as follows:
The figure is so called because the expression adds to the sense so much that it exaggerates it, and enlarges or diminishes it more than is really meant in fact. Or, when more is said than is meant to be literally understood, in order to heighten the sense.
It is the superlative degree applied to verbs and sentences and expressions or descriptions, rather than to mere adjectives. . . .
It was called by the Latins superlatio, a carrying beyond, an exaggerating.
The literary device of antithesis, or contrast also seems more specifically applicable to the verse we are considering. Bullinger writes about this:
A setting of one Phrase in Contrast with another.
. . . It is a figure by which two thoughts, ideas, or phrases, are set over one against the other, in order to make the contrast more striking, and thus to emphasize it. . . .
The two parts so placed are hence called in Greek antitheta, and in Latin opposite and contraposita. . . .
It is called also contentio: i.e., comparison, or contrast. When this contrast is made by affirmatives and negatives, it is called Enantiosis, see below. The Book of Proverbs so abounds in such Antitheses that we have not given any examples from it.
Jesus is expressing Hebrew hyperbole and/or antithesis to express with extreme exaggeration what he literally means: “does not esteem them less than me.” Thus, the thought of “loving Jesus more than one’s own family” is expressed by the non-literal “hate [one’s family, in order to] be my disciple.”
In fact, Jesus did express what we contend He was stating non-literally in Luke 14:26, in a literal fashion elsewhere (and here we see the important hermeneutical principle of “interpret less clear or obvious passages by more clear related passages”):
Matthew 10:37 He who loves father or mother more than me is not worthy of me; and he who loves son or daughter more than me is not worthy of me;
We see precisely the same parallelism (“hate” = “love relatively more than”) in the poetic literary expression of Genesis 29:30-33, which is a discussion about Jacob, Rachel, and Leah (“he loved Rachel more than Leah” = “Leah was hated”).
The apostle Paul expresses largely the same sort of thing in the same way:
Philippians 3:7-8 But whatever gain I had, I counted as loss for the sake of Christ.  Indeed I count everything as loss because of the surpassing worth of knowing Christ Jesus my Lord. For his sake I have suffered the loss of all things, and count them as refuse, in order that I may gain Christ
St. Paul casually alluded to apostles like himself and Peter (“Cephas”) having “the right to be accompanied by a wife” (1 Cor 9:5).
Jesus taught that we are to love (not hate) even our enemies:
Matthew 5:43-44 You have heard that it was said, `You shall love your neighbor and hate your enemy.’  But I say to you, Love your enemies and pray for those who persecute you, (cf. Lk 6:27-35)
Obviously, then, he would not (and did not) teach that we ought to hate our own families. Jesus taught that we should love all people, and that includes families:
Matthew 19:19 Honor your father and mother, and, You shall love your neighbor as yourself.
Matthew 22:37-39 And he said to him, “You shall love the Lord your God with all your heart, and with all your soul, and with all your mind.  This is the great and first commandment.  And a second is like it, You shall love your neighbor as yourself.”
Mark 12:31 The second is this, `You shall love your neighbor as yourself.’ There is no other commandment greater than these.”
Luke 10:27 And he answered, “You shall love the Lord your God with all your heart, and with all your soul, and with all your strength, and with all your mind; and your neighbor as yourself.
John 13:34-35 A new commandment I give to you, that you love one another; even as I have loved you, that you also love one another.  By this all men will know that you are my disciples, if you have love for one another.
John 15:17 This I command you, to love one another.
I rest my case. This passage (thrown in our face by many atheists) poses no problem whatsoever for either Christians, or a consistent interpretation of these Bible passages. It’s simply a function of non-literal forms of speaking that were common in Hebrew culture.
Not understanding this leads to innumerable misguided readings of Scripture from atheists and other biblical skeptics who appear to make little effort to try to comprehend it. They’re too busy tearing down Holy Scripture and approaching it like a butcher viewing a hog.