"The best is yet to be" means "things are good now, but the future will be even better."
It is a line from a famous poem by the great British poet, Robert Browning (1812 - 1889). You hear it in ordinary conversation, because the line is so famous that people quote it without even knowing that it is a quotation.
It sounds unnatural because it is nineteenth-century poetry, and what you have quoted is a rhyming couplet. The word "be" is perfectly correct here. However, I am sure that Browning chose "be," rather than "come," because he needed a word that rhymed with "me."
The poem is "Rabbi ben Ezra:"
The English is difficult even for native speakers. It is full of unusual words and unnatural word ordering. During Browning's lifetime, people formed clubs, the "Browning societies," and tried to understand the poems as if they were mystical puzzles.
The poem seems to be about thinking of a life as a whole unit or pattern. Instead of thinking of youth as the good part, and age as the bad part, think of them as two different pieces that fit together into a perfect whole.
"Grow old along with me!
The best is yet to be,
The last of life, for which the first was made:
Our times are in His hand
Who saith "A whole I planned,
Youth shows but half; trust God: see all, nor be afraid!''
For an example of very unnatural word order, consider this line from the same poem:
"Irks care the crop-full bird? Frets doubt the maw-crammed beast?"
It means "Does a bird worry, as long as its crop is full? Does a beast worry, as long as its mouth is full?"