The main softwares that graphic designers use right now are all created by Adobe. You can go to adobe.com and download several free trial outs of many programs for 30 days. I would suggest Photoshop & Illustrator. http://www.adobe.com/
You have to really use illustrator or another vector based program when making logos. It makes life so much easier in the future when it comes to resizing things
No question, Illustrator... well really any vector based program, Corel Draw would work too. Illustrator is the industry standard though.
I use photoshop, it's really cool once you get the hang of it, Illustrator is cool to, never used it thoe.
The first half decent free alternative to Illustrator that springs to mind is Inkscape. Essentially it is an Open Source vector graphics editor, with capabilities similar to Illustrator, CorelDraw, or Xara X, using the W3C standard Scalable Vector Graphics (SVG) file format. It is available for Linux, Windows and the Mac. I've heard quite good things about it. Definitely worth a shot.