I've used, and taught, both raster and vector programs, and I like to describe raster (Photoshop) as painting, and vector (Illustrator) as drawing. But really, that's just a start. Painting programs, like Photoshop, are easy to understand, but vector programs, like Illustrator, can be confusing. So, for Photoshop, I usually just tell people to go ahead and get started, and feel your way around, but for Illustrator, I have to do a bit of explaining first. Please bear with me.
Vector art is line art. When Illustrator was first invented, it really took the place of drawing with a pen. Over the years the software has been developed to do more stuff, but at its heart is the pen. If you're learning about vectors for the first time, use the pen tool. Yes, there are other shortcuts that you can use later on, but to understand what the program is for, start drawing with the pen. I call it a *lightbulb over the head moment* - when I see someone really understand vectors. Use the pen.
I use Adobe Illustrator mostly for my cartoons. I draw them in pencil, scan in the rough sketch, and then trace them in Illustrator. Before the invention of vector programs, this was called *inking*. Of course, before the invention of illustrator, it was very difficult to make changes to a drawing after it was inked. With Illustrator, and using best practice, it's easy, and a pleasure. Even after all these years I'm still delighted.
The Adobe Illustrator classes that I taught always emphasized best practice. That is, taking the time to set up the file correctly so that if (and it's usually when) the client needs to make changes, it can be done with a couple of mouse clicks. If the file is created correctly, that is, by someone who knows what vectors are, changes are easy. If not, it can be the worst snarl you've ever seen. I've had people send me files to make a small change that were so badly snarled it was actually easier, and took less time, to completely re-create it.
By the way, I did that cartoon there in vectors. In addition to creating a nice-looking cartoon line, which is easy to reshape, add colors, and change colors, it can be output to any size because there are no pixels. It's not raster, it's vector.