If your just getting into reading the four gospels, one thing to note, the originals were pretty worn, at times only fragments of a single sentence was recovered so the four gospels are not in chronological order at all more like torn into four pieces and shuffled. you mentioned the geno, here I have put together the verses into a more readable flow, if your interested. and yes 14 does have meaning.
Mt1:1
A record of the genealogy of Jesus Christ the son of David, the son of Abraham:
Mt1:17
Thus there were fourteen generations in all from Abraham to David,
Fourteen from David to the exile to Babylon, and fourteen from the exile to the Christ.
(14) ~ generations ~ Mt1:6b: (1st) David was the father of Solomon, whose mother had been Uriah's wife,
Mt1:7
(2nd) Solomon the father of Rehoboam,
(3rd) Rehoboam the father of Abijah,
(4th) Abijah the father of Asa,
Mt1:8
(5th) Asa the father of Jehoshaphat,
(6th) Jehoshaphat the father of Jehoram,
(7th) Jehoram the father of Uzziah,
Mt1:9
(8th) Uzziah the father of Jotham,
(9th) Jotham the father of Ahaz,
(10th) Ahaz the father of Hezekiah,
Mt1:10
(11th) Hezekiah the father of Manasseh,
(12th) Manasseh the father of Amon,
(13th) Amon the father of Josiah,
Mt1:11 and (14th) Josiah the father of (Jeconiah) and his brothers at the time of the exile to Babylon.
(14) ~ generations ~ Mt1:12
After the exile to Babylon:
(1st) Jeconiah was the father of Shealtiel,
(2nd) Shealtiel the father of Zerubbabel,
Mt1:13
(3rd) Zerubbabel the father of Abiud,
(4th) Abiud the father of Eliakim,
(5th) Eliakim the father of Azor,
Mt1:14
(6th) Azor the father of Zadok,
(7th) Zadok the father of Akim,
(8th) Akim the father of Eliud,
Mt1:15
(9th) Eliud the father of Eleazar,
(10th) Eleazar the father of Matthan,
(11th) Matthan the father of Jacob,
Mt1:16
and (12th) Jacob the father of (13th) Joseph,
the husband of Mary, of whom was born Jesus,
who is called (14th) Christ.
(40) ~ generations ~ Lk3:23
Now Jesus himself was about thirty years old, when he began his ministry. [Baptism]
He was the son, so it was thought, of Joseph, and the son of (40th) Heli, [Mary’s father]
Lk3:24
the son of (39th) Matthat,
the son of (38th) Levi,
the son of (37th) Melki,
the son of (36th) Jannai,
the son of (35th) Joseph,
Lk3:25
the son of (34th) Mattathias,
the son of (33rd) Amos,
the son of (32nd) Nahum,
the son of (31st) Esli,
the son of (30th) Naggai,
Lk3:26
the son of (29th) Maath,
the son of (28th) Mattathias,
the son of (27th) Semein,
the son of (26th) Josech,
the son of (25th) Joda,
Lk3:27
the son of (24th) Joanan,
the son of (23rd) Rhesa,
the son of (22nd) Zerubbabel,
the son of (21st) Shealtiel,
the son of (20th) Neri,
Lk3:28
the son of (19th) Melki,
the son of (18th) Addi,
the son of (17th) Cosam,
the son of (16th) Elmadam,
the son of (15th) Er,
Lk3:29
the son of (14th) Joshua,
the son of (13th) Eliezer,
the son of (12th) Jorim,
the son of (11th) Matthat,
the son of (10th) Levi,
Lk3:30
the son of (9th) Simeon,
the son of (8th) Judah,
the son of (7th) Joseph,
the son of (6th) Jonam,
the son of (5th) Eliakim,
Lk3:31
the son of (4th) Melea,
the son of (3rd) Menna,
the son of (2nd) Mattatha,
the son of (1st) Nathan,
the son of David,
(14) ~ generations
Mt1:6a: and (13th) Jesse the father of (14th) King David. Lk3:32a: the son of Jesse,
Mt1:5c: (12th) Obed the father of Jesse, Lk3:32b: the son of Obed,
Mt1:5b: (11th) Boaz the father of Obed, whose mother was Ruth, Lk3:32c: the son of Boaz,
Mt1:5a: (10th) Salmon the father of Boaz, whose mother was Rahab, Lk3:32d: the son of Salmon,
Mt1:4c: (9th) Nahshon the father of Salmon, Lk3:32e: the son of Nahshon,
Mt1:4b: (8th) Amminadab the father of Nahshon, Lk3:33a: the son of Amminadab,
Mt1:4a: (7th) Ram the father of Amminadab, Lk3:33b: the son of Ram,
Mt1:3c: (6th) Hezron the father of Ram, Lk3:33c: the son of Hezron,
Mt1:3b: (5th) Perez the father of Hezron, Lk3:33d: the son of Perez,
Mt1:3a: (4th) Judah the father of Perez and Zerah, whose mother was Tamar, Lk3:33e: the son of Judah,
Mt1:2c: (3rd) Jacob the father of Judah and his brothers, Lk3:34a: the son of Jacob,
Mt1:2b: (2nd) Isaac the father of Jacob, Lk3:34b: the son of Isaac,
Mt1:2a: (1st) Abraham was the father of Isaac, Lk3:34c: the son of Abraham,
(20) ~ generations
Lk3:34d: the son of (20th) Terah,
Lk3:34e: the son of (19th) Nahor,
Lk3:35
the son of (18th) Serug,
the son of (17th) Reu,
the son of (16th) Peleg,
the son of (15th) Eber,
the son of (14th) Shelah,
Lk3:36
the son of (13th) Cainan,
the son of (12th) Arphaxad,
the son of (11th) Shem,
the son of (10th) Noah,
the son of (9th) Lamech,
Lk3:37
the son of (8th) Methuselah,
the son of (7th) Enoch,
the son of (6th) Jared,
the son of (5th) Mahalalel,
the son of (4th) Kenan,
Lk3:38
the son of (3rd) Enosh,
the son of (2nd) Seth,
the son of (1st) Adam,
the son of God.