As you said, -ㅂ니다 gives an extra polite and formal feel, so this form is commonly used when you address an audience, such as on TV and radio, or dealing with customers you serve. In a normal situation, the -요 ending is much more common. It is practically as polite as -ㅂ니다 while being shorter and sounding softer and more friendly, so it is by far the preferred form between adults. But note that you don't have to just stick with one. Mixing these two polite endings are common and not unnatural (there are certain expressions that sound better with -ㅂ니다, such as 감사합니다). So if you're addressing someone for the first time and want to be a little formal, you might start the first sentence with -ㅂ니다 and continue with -요. But it depends a lot on the person's age. If both of you are young (up to college years), -ㅂ니다 might sound unnecessarily formal.

Your understaning is right, ㅂ니다 is more polite way than -요. If you say in not formal situation, you van use -요.

Agreed with the other posts. In short, you learn ㅂ니다 first just because it's the most polite way to speak in Korean. This usage is honestly too polite and I doubt you will encounter a situation when you'll need to use it any time soon (such as a speech contest, job interview, meeting someone of high status, etc) You'll most likely be using 요 a lot more for the foreseeable future