Begin with this one and follow Hornblower chronologically throughout his career. I have read the whole series more than once and loved it. Hornblower is a very appealing character--touchingly unsure of himself (while at the same time being tremendously capable on a ship); he is seasick at times and, if my memory serves me right, tone-deaf as well. The story has a way of drawing the reader right into the middle of the action.
Hornblower is said to be based on the life of Lord Nelson, though I do not know to what extent.
If you enjoy historical adventure the Hormblower series by C S Forester is fun. My father read a lot of Forester and handed them down to me and my son when he was a teenager. It had been many years since I read the series and decided to start from the beginning and read them again. Have read # 1 and wasn't disappointed.
For those fans of military SF who like David Weber's Honor Harrington series, this is the book that Weber credits as his inspiration for Honor's career. MR. MIDSHIPMAN HORNBLOWER depicts the earliest steps of a British officer's naval career, while the following books cover later years. Rousing good tales!
C.S. Forester's Horatio Hornblower character became arguably more famous than the real Admirals Nelson, Cook and Drake combined. The saga covers Hornblower's career through a dozen major campaigns during the Napoleonic wars.
The Hornblower series of books are just a wonderful kick to sit back and truly relax with. I get lost in them in a good way. None of them are my typical 'long' book, but each one just a delight of stories, mishaps and adventures as you want Horatio Hornblower 'grow' in his chosen profession. After I finish one I can't wait to read the next one. They are truly charming. If you love the Hornblower series, you will also love the 19 book series by Patrick O'Brian which detail the adventures of Captain Jack Aubrey and his onboard surgeon Dr. Maturin. WONDERFUL SERIES BOTH OF THEM. Don't think if you've read one series that you don't need to read the other. The main characters in each are very different and both series are top rate!