Yes, knowing the difference between "Sie" and "du" as well as conjugating verbs correctly to reflect the use of the right form is very important. Germans do take this seriously! You use "Sie" for everyone you don't know, you meet in public, and who is not a family member, friend, or child, so basically about 90% of the people you could possibly talk to throughout the day. You can use "du" with family members, friends, and when talking to children. Switching from "Sie" to "du" once you start having contact with someone on a regular basis, e.g. colleagues at work, is a delicate matter and requires some diplomacy. Because once you're on a "du" basis, you cannot go back to "Sie". This would be very awkward. Therefore, people take their time and are very careful. And if you use "du" without clarifying whether that's ok, many Germans would feel disrespected or insulted. Having said that, in our modern "cool Google culture world", some Germans tend to believe that using "du" for everyone shows how cool, relaxed, and open-minded they are. But this doesn't exactly go down well with everyone. So if you want as little trouble as possible, and if you want to be as respectful as possible, use "Sie" until some offers you to switch to "du". Usually the older/ more experienced person or person of higher hierarchy is supposed to offer it, and not the other way round.
