You're right, it's not easy when you don't know the terms and vocabulary.
You talk about transparency but in Illustrator, what you need is not called transparency, it's simply a white shape you need to "divide" (eg. slice and then erase)!
Imagine your vector shapes are like assembling pieces of paper together and then cutting out the parts you don't need with an exacto cutter. In this case the white is truly white, as much as a piece of paper can be white; you call it transparent because you background canvas is white too, but trace a black rectangle behind it and you see it's simply another white shape. Transparency is really when there's 0% solid color or content.
To remove these extra part you don't need, have a look at the link I added to my previous comment about the "pathfinder" tool. With that tool, you can "merge", "slice", and do similar things you'd do in real with that kind of shapes if it was paper. That's how you'll get rid of the white areas in your artwork. With images, you'd probably use what's called a "clipping mask" instead.
You'll find that tool in the "Window" Menu in Adobe Illustrator.
Clipping Mask, Pathfinder or Compound Path?
It's very easy and will be very useful as well to learn about it for your future projects. In fact, you'll use this all the time. Note that if you use strokes, you need to "expand" them first (menu "objects", then "expand")