I am just starting to teach myself how to make logos. If I want to make a logo that works with both Apple and Android apps, what software do you recommend that would help me do a logo that would be compatible on Apple and Android devices?
To give you the answer you're looking for, most people would probably still be using Adobe Illustrator to design logos. There are some free alternatives to that and of course, Zach's answer should also be considered here as the software will not design logos by itself.
Many times a client will not know what kind of logo to ask for, so you need to be able to work with limited or vague briefing, have a good understanding of composition and spacing and be able to put things together visually before you get to the software part.
Whatever you do, make sure you use a vector-based software so the resulting artwork will allow exporting to different formats and sizes (iOS/Android apps, web, print, etc).
I recommend using pen and paper. Sketch a bunch of ideas. Take about 3 that you like and try a bunch of variations of them. Eventually you should get something usable.
Then after you get something that you like, you can use whatever software you want to create a digital version of it. If you need it in vector (it's probably the most useful in vector form), Adobe Illustrator or Inkscape are your best bets. I'd likely use one of them. If you know ahead of time that you only need raster images, you can use a plethora of things, but Gimp is a free tool that is good.
When it comes to images on mobile platforms, it doesn't really matter what software you use to create it. An image is an image, both can render it. Thus all you need to consider is what software you would prefer to make the image.
Adobe Illustrator perhaps works best for me. It is a vector-based program that renders graphics with clean lines and edges, and allows scalability without losing quality. Plus there are lot of online logo tools that let you make logos in minutes. I have used one, DesignEvo. Decent app, though its features are a little basic in some way, it is still a not bad choice to play around with. Hope this can help you.