I have the first 4 or 5 book in the series . The first few were good, well written works but I've not yet read beyond that.
I liked the story when it first came out as it was a little different combining as it did magic, dragons in the form of an 'air force' and military history in the guise of the Napoleonic wars. Problem was/is the books are a little too much the same, that is the premise and the style in which the books were written wore a little thin for me. Having said that they are still enjoyable reads and I would go with the order of the books you have eventhough they are intended to be stand alone stories per se despite having same central characters. I think you definitely would want to read book 1 as it introduces the main characters...beyond that it's not such an issue.
Following is a very handy summary (without obvious spoilers or fairly minor/broad at least) lifted from Wiki (apparently intended to be 9 books in the series):
The series of books revolves around the primary characters Temeraire and Captain William Laurence. Captain Laurence is a member of the British Royal Navy, serving in combat against Napoleon's navy when he recovers a dragon egg unlike any other known to the British. The egg soon hatches, and Temeraire, a Chinese dragon, is born. Under the impression that an "unharnessed" dragon will become feral and unmanageable, Laurence becomes Temeraire's companion. Despite the difficulties this causes, Laurence begins to think of the dragon as his dearest friend. This forces a change in the sailor's life, drawing him from the prestigious Royal Navy to the less desirable Royal Aerial Corps. The subsequent novels in the original trilogy follow the adventures of Laurence and Temeraire as they do battle with the forces of Napoleon Bonaparte and the diplomatic fallout caused by Captain Laurence's adoption by the Chinese Emperor. The fourth novel deals with Laurence and Temeraire seeking a cure for a draconic illness, introduced by a North American dragon, which spreads throughout the British dragons while Napoleon seeks to press his advantage. The fifth novel is the account of Napoleon's invasion of England, forcing a British retreat to Scotland, while Laurence faces with the consequences of his treason in taking the cure for the illness to the French. The sixth novel begins within the penal colony of Australia (Laurence's death sentence for treason commuted to transport to the colony), and a chase across the continent to a discovery that has far reaching consequences for the global war
I recall meeting Naomi (and her husband) in '07 at WorldCon when she was first starting out. It was the same event that saw her receive the John W. Campbell award for best new writer.