Alright,i want to learn. :redface:
Possibilities are really endless. There's a lot of software available now specially for graphics, ranging from things like Paint.NET and GIMP to Photoshop and Corel.
You don't need the "best" or the most expensive to get some good designs, but they obviously have a larger selection of tools and features. Some of them can be quite difficult to get to grips with, and make take some learning.
It really depends on what you wanted to do. I personally make use of several different graphics/presentation software to try and achieve the exact things that I'm trying to create. Sometimes I use different graphics packages for each element of a logo.
Complex? It doesn't have to be. Professional graphics designers are obviously quite skilled and their resources are likely to be a lot better than what we have at hand. But even things like Microsoft Powerpoint for instance has a number of useful tools. As it's mainly for presentation purposes some of its features can be useful even for creating graphical pieces.
As an example, the 2 logos below - I created them recently for parts of the ARPD, VCPD:

Though I used more than one program for these, I began by putting some of very general 'starting pieces' together on Powerpoint. You obviously don't have to use Powerpoint, I've found other programs to be equally user friendly, but thought I'd talk about that for this example as it's a likely one for most people to have access to. It's very good as it's convenient to move things around, place them, generally plan things out. Also, you can actually change the size and type of the plate - the "page" or the slide, the bit you normally work on.
I'd recommend you give that a shot. Some of the tools on there may be useful, things like crop, removing background, changing sharpness, contrast, colour types, shadowing, glow... all of those may have little roles here and there. Of course, actual graphics packages may provide all of these features and more to a higher level of quality, but to begin with it's no harm in playing around with what is easily available.
It is a good place as any to start.
Creating an image right from scratch is time taking and can be challenging in places. Maybe you want to take parts of images from open source areas and begin by editing them to try and get them closer to what you want. Many of the programs out there actually have an image library or similar, where you can look up shapes and modify them to your preference.
When it comes to the actual designing also remember to plan if you had something in your mind that was very specific. Draw it out on a piece of paper, doesn't have to be detailed at all. It just gives you a benchmark to work by. You can add/change as you go along, and there's no harm as it is your own piece of work, it's your design and it wouldn't be "wrong" as such. Almost always I find myself adding things even after I feel I've finished, or changing minute details to get it more to how I (or the person I'm making it for) would like it to look. The logos above, I added the 'VCPD' lettering and the small wings completely unintentionally at first. I was just playing around with some extra shapes that I had and pretty much decided to include them at the very end.
It's a lot about the aesthetics and the way you get there shouldn't matter so long as it is your design and you're happy with it. Without that thought in mind it's easy enough to give up or get frustrated. It's very difficult if you don't enjoy it.
I have just talked about some general things really, I'm not sure how much use it was, but it can be harder to explain things like this through letters and words. :P
Maybe you could look it up as there are plenty of guides/blogs/videos that can be helpful. I'd first experiment for yourself.