If you're just getting started - don't worry, you don't need everything listed here! At minimum, you need photo editing software (Photoshop / Inkscape / GIMP) and a tool to convert planetary imaging data into a format editable by that software (IMG2PNG, which is Windows-only; if you know a good option for Mac/Linux, let me know). I am mostly a Windows user, so most of my toolbox is Windows oriented. I have recently moved to a dual boot setup with Mint Linux which enables a few advanced processing options.
The most essential tool for image processing on Windows is Björn Jónsson's IMG2PNG utility. This command line tool quickly and easily converts images stored in NASA's PDS archival format (.IMG) into a lossless image format that is readable/editable by photo editing software. Almost every image I have edited has passed through this utility at some point or another.
My main workhorse is Adobe Photoshop CC. I used to work in Photoshop CS5, but the disk has since gone missing and I was no longer able to bring it with me when refreshing/upgrading computers. I hate subscription software and am slowly moving away from it, but it's what I grew up using and everything else feels clunky in comparison. A lot of my workflow has moved over to GIMP, the GNU Image Manipulation Program. This is an open-source photo editing suite with a lot of features that rival Photoshops, especially with the G'MIC plug-in, which provides a powerful set of tools for noise reduction, image sharpening, and contrast enhancement. In fact, much of my workflow now uses GIMP, but there are some clunky elements to the base program that prevent me from abandoning Photoshop completely.
ImageJ is a Java-based "swiss army knife" for image processing. It's something I rarely use these days (most of the filtering functions have been supplanted by G'MIC), but it is still a core piece of software for certain types of image sharpening.
Since moving to a dual-boot setup with Linux, I have started incorporating USGS Astrogeology's Integrated Software for Imagers and Spectrometers (ISIS3) into my work flow. This program is designed for planetary cartography, and is can interact with imaging data at a much lower level than IMG2PNG. Unfortunately, it doesn't work on Windows. In years past, installation used to be a major hassle, but recently the USGS has been working to make the software more user-friendly. I will probably develop some tutorials for using it as time goes on.
Finally, in the last year I've been developing my coding skills using Python 3. Much of this was spurred by a need to move past ASU's Davinci software for processing THEMIS data at work. It has now been nearly a decade since the last update and is beginning to prevent upgrades to other software. As my skillset matures and I develop some useful toolsets I will add a GitHub page.