It's an action-RPG. You're trying to avenge your father's death. Along the way, you're finding clues while exploring an immensely detailed world. Everything about the world is painstakingly crafted: NPCs all have set schedules, there's a day/night cycle that they actually follow, interiors are all custom-crafted. There's an authentic arcade you can visit filled with old Sega games. It's about as close to exploring 1980s Japan as you can get. This might sound like a lot of open-world games today, but the way it's all brought together is truly unlike anything else.
You can also do side jobs, like driving a forklift or running a gambling stand.
When it comes time to fight, you're basically playing Virtua Fighter-lite. There are special moves to learn through training. It's a really fun combat system and unlike other RPGs.
What's really special about Shenmue is the minutiae. You can interact with all sorts of things around you, even if it's bullshit like picking up a cup and spinning it around in your hand or taking a picture off the wall. There's this incredible sense of presence that you just don't get from other games.