The first Vorkosigan book in publication order is Shards of Honor. Falling Free takes place a couple of hundred years before the rest of the books in the series, so while technically first chronologically it's more of a stand alone book.
Normally I recommend folks read series in publication order, and you could certainly do that here, but in the case of the Vorkosigan series you probably want to read them (more or less) by internal chronology. I recently turned a friend of mine onto the books; here's the order she's read them in so far:
1. Shards of Honor
2. Barrayar
(These are available as the omnibus
Cordelia's Honor.)
3. The Warrior's Apprentice
4. The Vor Game
5. The Borders of Infinity (containing "The Mountains of Mourning", "Labyrinth", and "The Borders of Infinity".)
(The first two and "The Mountains of Mourning" are included in the second omnibus volume,
Young Miles.)
6. Cetaganda
7. Ethan of Athos
8. Falling Free
(The first two and "Labyrinth" are included in the third omnibus,
Miles, Mystery & Mayhem. Falling Free, as mentioned above, stands alone completely and can be read at basically any time.)
9. Brothers in Arms
10. Mirror Dance
11. Memory
(The first two and "Borders of Infinity" are included in the fourth omnibus,
Miles Errant. Memory is only available as a stand alone book. Brothers in Arms is the point at which you really want to make sure you're reading the series in chronological order.)
Things progress from there. You can see another reading order recommendation
here or check out the
Wikipedia entry for the Vorkosigan series and just follow the publication order.
tl;dr = Start with
Cordelia's Honor (Shards of Honor/Barrayar).
FnordChan