As you've already heard, Illustrator is a good option as long as you can afford it. If not, the free option would be something like Inkscape. I would recommend that if you don't know how to use vector design software, start with paper and scan or use a digitising tablet.

As mentioned, look for tutorials online depending on the style you want to achieve. The disadvantage of a vector programme like illustrator or inkscape, is that when you are a beginner, all the drawings you do there are still going to look like balls and sticks.

Illustrator, Adobe online has several free tutorials where you can learn the basics to create your characters :D !