The OP specified that he wanted an easy-to-learn software. Illustrator has horrible tutorials and a horrible learning curve. Even for formally trained designers who didn't buy every upgrade since Illustrator 10, the software is difficult to use. Illustrator, while the industry standard, is not the answer to the OP's question.
Depending on what your goals are, one can design a decent logo in the Microsoft Publisher program already on many office computers. It will let you save your image as a 300dpi jpg or png. If a client brings me a basic logo they made themselves at 8.5"x11" 300 dpi, I can live trace it for them and touch it up for other uses.
Only professional designers should invest the time and frustration needed to get used to Illustrator. Let's face it, Adobe isn't what it used to be. Each "upgrade" is buggier and more problematic to use than the last.