How to make this vector animation?
That is the realm of motion graphics and is probably using vector graphics within the motion graphics.  To get something of that quality, would take a very long time for a beginner to make look that good, though if someone is naturally skilled at it, it could shorten the curve considerably.  It requires skill in multiple disciplines to produces a reasonably high production value video like that.

If you want to try to pursue it though, I would suggest looking at software like Flash or After Effects.  Illustrator is also worth looking at for developing the vector assets.  After Effects with Illustrator is probably the best bet as it supports 3d layers (which could be used for the 3D spinning) as well as fairly easy support for write on text.  The painting effect is probably a little easier in Flash, but should be plenty doable with some work in After Effects.

Your early work isn't going to look anything like the video you linked though.  If you want such a video for commercial purposes, your best bet is to pay a studio to do something for you.  If you just want to play around, then dig in to the topic of Motion Graphics which is way too big of a topic to cover in one question and answer.