The industry standard programs are Adobe Illustrator or CorelDraw for vector graphics and Adobe Photoshop for raster graphics. Popular alternatives are Inkscape and Gimp, I believe both are free.
If you are working with a budget that easily allows you to trademark your logo, I would highly recommend it. But if you are working with a tight budget, I would highly recommend spending your money on production and marketing. Leave the legal stuff for later on when you can afford it.