Why everybody forgets CorelDraw?
It is a FAR more easier and simpler for anyone to learn it, than illustrator's intuitive and complicated interface.
And it has the same strength as illustrator. I even don't use Layers at all. I work very freely.
I am a graphic designer and I use CorelDraw for many years now. The Latest version is X6.
By the way, CorelDraw is not Free, the same as illustrator. But Both Softwares are for professional graphic designers.
=> But in your case now you will need this software "AAA Logo Creator" which provides you with many symbols and cliparts to use and it is very easy to use. This is the software that "Aneekalam" is using and provides their services in Blackhatworld forum, as paid services. Their Thread has this title: (Stunning Graphics Designing! Logo,Banner,Flyer,Landing Page,Infographics Starts from 19$).
=> A Free and alternative to corel and illustrator is "Inkscape", which is a vector graphics software as well.