To start with, if you haven't yet read Shards of Honor, the first novel in the tales of Miles Vorkosigan (even though it doesn't actually feature Miles, it's about how his parents met), STOP, go back and read it before approaching Barrayar. You'll be glad you did. These two novels, written seven years apart, tell one complete story arc. How that came to be is an interesting story.
Shards of Honor and Barrayar form the beginning of the Miles Vorkosigan series. (Bujold's novel Falling Free takes place within the same fictional universe but, being set approximately 200 years before Miles' birth, features none of the series' familiar characters. Eventually you'll want to read Falling Free, but it doesn't matter when; you can insert it into your Bujold reading experience anytime.) Shards of Honor is Bujold's first novel (not merely the first novel she ever sold, but the first she ever wrote, thus disproving the axiom, "All first novels are unsaleable trash"). She begins writing it in December 1982. In mid-'83, having worked through the Shards material and about a third into what would eventually become Barrayar, Bujold realizes her manuscript is becoming too long to submit as one book (the "wisdom" at the time being a thin manuscript is more likely to be picked off the slush pile than a thick one). Bujold finds a logical breaking point for her tale (Cordelia's arrival on Barrayar), puts it in final draft form, and mothballs the partially finished "rest of the story."
Bujold submits Shards and begins working on another book, The Warrior's Apprentice. She's about halfway through that when Shards comes back rejected with an editorial suggestion she tighten it up. She finishes Warrior's, then cuts about 80 pages out of Shards, giving her two good unpublished novels. In 1985, around the time she finishes her third novel, Ethan of Athos, Warrior's makes it over the transom at Baen, and suddenly she goes from unpublished wannabe to successful novelist with three books (Shards, Warrior's, Ethan) SOLD. Shards is published in 1986.
Fastforward to 1989. Bujold has written four more books, Falling Free, Brothers In Arms, Borders of Infinity (a short story collection), and The Vor Game. Then the program-book editor of Philcon, a long-established SF convention in Philadelphia, asks Bujold to do a short story or outtake to donate to their program book. Remembering the unfinished novel fragment of years before, she troops up into her attic, retrieves the pages, reads them and decides to complete it as a novel. After all, it's already a third finished, right? And in 1992, Barrayar won the Hugo Award for Best SF Novel of the Year.
Shards of Honor stars Captain Cordelia Naismith, commander of a survey team for the Betan Expeditionary Force, and Captain Aral Vorkosigan, victim of a mutiny on his Barrayaran warship. Both stranded on an unexplored alien planet, officers on opposite sides of the Betan-Barrayaran War, they reach an agreement of honor: they will trust and rely on each other for survival as they travel across a planet seemingly intent on throwing all its resources into killing them before they can reach Aral's ship. And then there's the little problem of overcoming the mutineers.... In the process of their adventures, Cordelia and Aral fall in love.
Barrayar deals with her first experiences on that planet, leading up to the birth of her and Aral's son Miles (though there is an epilogue showing Miles at age five). Both Shards and Barrayar are told from Cordelia's perspective. Thereafter in this series Miles, with very few exceptions, takes center stage. Never again will Cordelia be the main character. But for these two books she emerges as one of the most well-realized, loving and vulnerable but still tough-as-nails female SF protagonists ever.
The next book in the series you'll want to read is The Warrior's Apprentice, which picks up Miles' life at age 17.