This is easier than Tobira, and contains many more explanations in English. I think the sections on Japanese culture explained in English are very insightful too. Harder vocabulary in AIAtIJ only starts turning up in later chapters. The first chapter just feels like a dialogue for beginners with keigo added. Very accessible. QUARTET is from the same publisher (The Japan Times) and is much more recent. It might be more suited to Japanese as it’s used today. I haven’t looked at it much though. Perhaps you should check it out.
The textbook I have personally is Tobira, but I’m about to drop it because I’ve been able to learn almost all the grammar inside it from anime and dictionary/Google searches before even touching the lessons inside the book. (I’m at chapter 13.) Tobira does go into some depth about Japanese society and culture, but I feel it can get rather dry at points if the subjects chosen don’t interest you. Disclaimer: I’m a university student, so perhaps I subconsciously prefer stereotypical ‘university’ topics like ‘how to handle a homestay’. Who knows? Either way, it’s starting to bore me. That doesn’t mean it’s not a good textbook though! It’s just not feeding me enough new information per chapter for it to be worth my time.
Ultimately, I’d suggest you take a look at the textbooks and workbooks on the list that @Kraits posted (except maybe the ones whose names are entirely in Japanese, because they’re definitely too advanced for you if you find the first chapter of Tobira difficult) and see which one has a style that best suits you. Tobira may be vaunted as the ‘best’ because it probably has the most technical terms and complex vocabulary (along with being more interesting for adult learners who aren’t university students), but that doesn’t mean it’s the best for you. Look for something that suits your needs and allows you to progress. All the best!
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