Japan not only has a long practice of making their own Space Operas, they're also very practiced as franchising the hell out of them, such that Lucas's building up such a business with Star Wars can look shabby by comparison. There is no better example of a long-running Japanese Space Opera franchise than the Macross franchise, and it all begins with the '80s classic that remains a great work to itself: Super-Dimensional Fortress Macross.
It's huge, with entries going well into the present (Macross Delta is the most recent), but you don't have to take all of that in at once. The heart of the series, and the franchise, remains the original early-80s series. Start here if you have no familiarity with the franchise. You swap lightsabers and explicit superpowers for fighters that transform into giant robots in order to fight giant aliens in giant robots, and the importance of culture towards victory.
Our hero is the typical Japanese protagonist: a young, earnest man out to prove himself. The villains are not all unredeemable, but the ones that are most certainly are properly villainous and come to satisfying ends. The heroines are feminine, and the villainesses by their lack of it. Victory comes bittersweet, very much so (in typical Japanese style), but it is a victory in both the military and the moral sense.
"But wait! This isn't interstellar!" you say. Nonsense; it's "Space Opera", not "Star Opera", and man are the relationships in this series both true to life and tense at times. The heroes face challenges that they don't always overcome, and when they fail you can see how their flaws made those failures possible. The music that this series is famous for is not just a soundtrack; it's a key element in many stories in the franchise, often plot-defining ones, and the relationships are signified by the music featured.
If you love action, you won't be disappointed. While there isn't much in the way of fleet battles, this series is famous for its dogfights and man-to-man action scenes. You'll get that in spades here, and you'll get them with heart, tension, and occasional tragedy. While other entries may do this or that better, the original series has it all; you won't be disappointed.
A note about the rest of the franchise: Other entries are not strictly Space Opera. Macross Zero is entirely on Earth, as it's a Prequel to the original series, and more of Weird War with the usual Romance. Macross Plus is far heavier on the Opera than the Space, as it's about dueling test pilots and the woman in common between them. Macross 7 gets a bit wacky at times, mocking its own tropes at times. Do You Remember Love and Macross II are in-setting feature films. Frontier and Delta are (like 7) proper sequels in the spirit of the original.
As such, many of the issues we talk about with Star Wars have come up here, but never has this franchise suffered from the same problems that the Mouse did to Lucasfilm. Nonethless, it's worthwhile to see where they misstepped, and how they dealt with it.