Try James Alan Gardner’s League of People books.
The basic premise is that humans are by far not the biggest kids on the block and there is a universal proscription against killing sentient beings and leaving a given star system. Each book in the series is told in the first person, with the main protagonist being a new character who may or may not interact with the protagonist in the first book. Often the book is focused on solving a mystery. While the books can probably be read in any order, the little details and background bits kind of accumulate more pleasingly if your read them in publication order.
The Books are: Expendable, Commitment Hour, Vigilant, Hunted, Ascending, Trapped, and Radiant. Vigilant and Trapped are probably my favorites.
Lois McMaster Bujold’s Vorkosigan Saga is fantastic (I enjoy the Cordelia books and the stories in Mountains of mourning, but my favorite books are Memory and later)
I’ll second David Weber’s Honor Harrington Series, also agree with the need for an editor as the series progresses. You can read the first book (On Basilisk Station) for free at the Baen Free Library (and you can get pretty much the entire series electronically on one of the later Baen Free CDs).
If a little fantasy’s more your thing, you can try his “Lay of Bahzell Bloodyhand”: Oath of Swords, War God’s Own, and Wind Rider’s Oath.
A shorter intro to Weber might be his Empire from the Ashes (omnibus of three books), or In Fury Born (essentially a re-release of Path of the Fury with a novel length prequel).
All of these are available free on Baen CDs
Heinlein is almost always worth reading, either in short story, juvenile or later novels.
Edgar Rice Burroughs was my introduction to science fiction/fantasy, especially the Martian Books.
I’ll also second (third?, fourth?) the first three Dune books (especially Dune and Children of Dune) . God Emperor of Dune was such a slog, and I’m not sure that the payoffs in the later books are worth that effort.
John Varley’s “Steel Beach” and “The Golden Globe” are pretty solid reads, and I’m sure there are other Varley fans out there with further recommendations.
One last recommendation I’ll make here (although I could go on, I had a roommate with well over 5000 sci-fi/fantasy books and she introduced me to many new authors) would be Tanya Huff’s Valor Series: Humans are one of three species recruited to fight a war that older, more civilized species cannot handle. One of the other species is insatiable and indiscriminate when it comes to sex, the other is arboreal and aggressively omnivorous.
The first book, “Valor’s Choice,” has our heroine lead a platoon on a ceremonial/diplomatic mission to recruit a fourth new species. Things go horribly wrong.
The second book, “The better part of Valor,” has our heroine investigate an apparently derelict alien ship discovered by accident. She is saddled with a scratch built platoon, and egomaniacal CO, and a stubborn, agoraphobic Salvage Operator. The annoying reporter is the icing on the cake. Things go horribly wrong.
The third book, “The Heart of Valor,” has our heroine accompany a senior officer recovering from battle wounds as he observes final training exercises for a platoon of new recruits. Things go, you guessed it, horribly wrong.
The last book, “Valor’s Trial,” has things go more wrong than usual in that everyone thinks our heroine is dead after a horrific battle. Our heroine wakes up in an apparent POW camp and sets about, once again, kicking ass and taking names.
-DF