Odds are you can not successfully use computer vectorisation tools on any but the simplest of bitmap images. For any but lowest of low quality logo work the only real option is to draw it from scratch.
Drawing is illsutrator is a much more "left-brain" thinking of raw reasoning, than the artistic feeling you get by painting. The key to illustration is planning and precision. As a reward you get a drawing that is incredibly easy to change later for new uses. Something that painting has no match for.
How to learn Illustrator
Start with simple things and take your time mastering those simple things.
The first tool you should master (sic) is the line tool. No, not the pen tool or any other tool the line tool. The line tool is first and foremost the tool you need to build the scaffolding to get your perfect shapes.
The first lesson of the line tool is to understand that accurate placement needs grid snapping, smart guides and typing in the creation box by alt clicking on your canvas.
TIP: use round joins and line caps at first less things to worry about.
Second tool you need to learn is the direct selection tool. This tool lets you modify your designs. I find that this is by far the most used tool in Illustrator but YMMV.
The lesson to learn with direct selection tool (white arrow) is to understand the interplay of shift, control and alt after you started dragging (pressing them before has a different meaning). And then understanding where you start your drag affects what point is going to snap.
The next tools to learn are the rotate, scale, shear and reflect tools. Mostly in that order. While it may seem that you can do these operations with selection tool, that is simply not true.
Again the key is to discover again that clicking once allows you to move the pivot where you want it to and alt lets you type in a exact value. This is a bit obscure, but very important. A good 10-15% of all the questions of this board could be solved by having known the basics of these tools.
After you have familiarized* the line tools and can print dotted accurate lines, build scaffolds, and measured lines you can move over to making shapes. See illustrator primitives consist of strokes and fills.
To understand shapes think of a illustrator as being composed of pieces of paper cutouts laid on top of each other. The stroke is like drawing with a mechanical pencil on paper, while the fill is a paper cutout in the shape of those mechanical pencil lines. The paper piece can have a single color, it can have a gradient or it can have a texture. By laying these papers and manipulating them you end up with any shape you wish to use.
The first shape tool you should master is the ellipse tool. In fact i hardly ever need the other tools in the shapes dialog but then I do very specialized work. Again learn what using shift and alt does when using the tool. The first lesson of the day would be to color your circle transparent.
Invariably the next thing to learn is how to combine line work and circles into more complex shapes. To do this learn to use Shape Builder, join (ctrl+j) than later Pathfinder and cut tool.
Once you reach this point your finally ready to start master the pen tool! Now the pen tool is very complex and can do all shapes you would ever need. But its very non-intuitive the first time you do anything. Its best if you find a few tutorials on the subject.
Remember the lessons you learned about doing lines with snap and smart guides, now use it for your advantage. You can also use line intersections as guides for where to place your control points and handles. By the time you reach this a few weeks should have passed and you should have picked up quite a few things.
You have now reached a point where you should be able to do most things you need to do outside typography (which you should be somewhat familiar with). Rest of the tools should come in time. When you encounter a specific problem than the use of other parts of the software should become more apparent.
There are no shortcuts
* Mastery of these tools would take years since its not so much about the tool but rather the problem at hand.