Museum Altes Schloss in Stuttgart will teach you about the state's history, museum is rather newly updated and offers good all-around knowledge and lesser known facts (e.g. relations with Russia where quite some progressive queen's came from, architecture with riding stairs,...) Grabkapelle Württemberg, also Stuttgart, Mausoleum of royals with interesting back story and beautiful echo effects in the crypt, built on the ruins of the former home of the Württemberg kings. Schloss Solitude, Stuttgart, beautiful view, hunting castle of kings with amazing anecdotes you'll get to know during a tour. Taking a guided tour through Stuttgart (often at night) is also amazing.

You can also visit 2 remaining WW2 bomb shelters in Stuttgart which are pretty neat.

That's what I'd recommend in Stuttgart, otherwise do visit further south.

Kloster Lorch, about 20 minutes out of Stuttgart, is also amazing. Has a huge mural inside depicting the complete history of the house of Staufen 😍

Especially if you don't speak German, I'd recommend guided tours and audio guides in English. They will tell you a lot about historical events and anecdotes and it must have been quite a peculiar state back under royal rule.