The first thing you should do is to get out a piece of paper and pencil, and make some sketches of potential designs for your logo. You need to get a good solid idea first. Then, you will finish it on the computer. That is easily done. You can take a digital photo of your final sketch and use it in whatever graphics program you use. But, you need a good idea. I'll bet you have some ideas floating around in your head right now. So, you have to get them down on paper. It is a lot faster to sketch out an idea on paper than to use a graphics program. (The only exception to this is to use a graphics tablet hooked up to your computer. But, you said you were on a budget.)
If you need feedback on your initial designs you could post them to Deviant Art or find a good graphics design Group on Flickr. A lot of artists use Flickr to share their work.
Inkscape is a good program. I learned to use Illustrator extensively in some college course some years ago. I could never afford a copy for my own use. I tried Inkscape and was very happy with it. You cannot beat the Pen Tool in these two applications for making good solid sharp linework. Inkscape has a Pen Tool and most of the other features that Illustrator has that you would need to produce a logo. There are tutorials available online for Inkscape.
I have heard that Gimp is a good program. I use Paint.Net. If your logo is simple enough you may be able to do a good version of it using the graphics in MS Word. I am not familiar with the graphics available in PowerPoint.
Best of luck