It's important to know how to address people. In the following example, an interviewer asks the host of the fashion show, Floria, Princess of Hesse, how she is to be properly addressed:
...also das ganz Korrekte ist „Königliche Hoheit".
...that is, the really correct [way] is "Royal Highness".
Caption 8, Modedesignerin Nina Hollein - Floria Prinzessin von HessenPlay Caption
Germans have a reputation for formal address. Traditionally, even co-workers of many years always addressed each other in the formal second person (Sie) and used both formal titles (Herr Doktor) and the person's last name. Anyone who has watched The Sound of Music knows that young ladies are to be addressed as Fräulein.
All of these things have changed a great deal in the past twenty years, and it is now much more common to address even a band you've never met before, as in the following example, in the informal second person (du or ihr, not Sie), or at least only use their first and last name without a formal title.
Und, äh, ihr habt ja auch was mitgebracht...
And, uh, you have also brought something along...
Caption 8, Undertube - Jeans TeamPlay Caption
The way of distinguishing between married and unmarried young women has all but disappeared, so if you meet a woman for the first time it is always appropriate to address her with Frau and her last name. Even Princess Floria concludes her thoughts on the topic, which began this lesson, by saying that the formal mode of address would be inappropriate in this context.
...aber das ist, glaub' ich, etwas fehl am Platz.
...but that is, I believe, somewhat out of place.
Caption 9, Modedesignerin Nina Hollein - Floria Prinzessin von HessenPlay Caption
It is still most polite to wait to be offered the informal (du) if the person you are meeting is your senior or in a position of authority.
Remember that while Sie sounds like the third person feminine or third person plural (sie), it is distinguished from them by always being capitalized. Also, it is conjugated like the third person plural (Sie/sie haben), not the second or third person singular (du hast or sie hat).
When learning a new vocabulary noun, memorize the definite article (der, die, das) along with the vocabulary word itself. That way you won't have to figure out what the grammatical gender is when it comes time to use it.