It's possible to read each sonnet as a stand-alone poem, but they are richest when read as a story. I wouldn't say that they must be read in order from start to finish, but to me a single sonnet, without its larger context in the sequence, is a petal without a rose.
While Shakespeare didn't bother to compile and publish his plays in his lifetime (his buddies did it after he died) his sonnets were compiled and published in 1609. It stands to reason that it was important to him they be published, and that he would have presided in some way over that process.
Before the 1609 edition, a few sonnets appeared in literary collections around London alongside poems by other, contemporary authors. But other than these few exceptions, the sonnets were private poems for many years, circulating only amongst Shakespeare's personal and professional friends.
The sonnets have garnered a popular reputation as a collection of sickly sweet love poems. A gold-gilt copy of the sonnets is a common wedding gift for newly married couples. But these are not exactly poems about conjugal bliss.
There's some evidence that these poems' sugary reputation isn't new. In 1598 a woman named Francis Meres referred to them in a letter as Shakespeare's "sugred sonnets." I wonder which sonnets Francis was reading.
Though it's not untrue that many of the sonnets are sweet, they're often sweet in unintuitive and even deeply weird ways. Sonnet one serves perfectly as an example.
The first and majority of the poems address a beautiful young man, often referred to as the "fair youth" sonnets. This relationship eventually breaks off, for reasons we'll get into later, and the poet gets involved with a woman. Traditionally poems that have to do with her have been called the "dark lady" sonnets. Finally, in one of the more ridiculous plot twists in literature, the boy from the fair youth sonnets shows up again - this time in the poet's relationship with the woman.
All kinds of questions have been raised about this story. Could this have actually happened in Shakespeare's real life? What is the identity of the boy? Who was the woman? I think these questions can be fun to consider (up to a point) but ultimately they're fruitless. Really, nobody knows.
Personally, I've always liked to think Shakespeare's "fair youth" could have been the boy actor who played Lady Macbeth, or Cleopatra. I share this to make the point that my pet idea is as baseless as any of the hundreds of theories people have thought up. James Joyce wrote that Shakespeare is "the happy hunting ground of those who've lost their reason," and truer words were never writ.
Could this story have happened in Shakespeare's life? Sure. In fact I think it's unlikely that the story is entirely made up. It's probably based on true events, or, like all literature, based on the experiences (real or imagined) of the author and people they knew. All that matters now is what we can take away from these poems.
Shakespeare's Sonnets, 1609 Quarto.