Probably the easiest tool to use is Inkscape. There is a learning curve to all software, though. Install it by typing the following command in a terminal:
sudo apt-get install inkscape
Inkscape is free. It is an open source equivalent for Adobe Illustrator. It allows you to draw vectors (stars, squares, cricles,..), to select their colors, to cut and paste them, to add text, and more.
Once you have designed your logo you can export it as a .png (pronounced ping) file. That file you can use in your website. If it is too large you can downscale it or you can make it a .jpeg file by opening it with gimp and saving it as a .jpeg file.