DESIGN SKILLS:
For starters, study the portfolio of Chermayeff & Geismar & Haviv to pick up some good design habits::
Domestika offers a very good, drirt-cheap logo design course with Sagi Haviv:
SOFTWARE:
Logos are desgined with vector graphics editors, to make sure they can be made as large as necessary. This way they can be upscaled indefinitely without any loss in quality. Vector graphics editors include Adobe Illustrator - the industry standard - Inkscape, Affinity Designer and Corel Draw.
Never create a logo using a bitmap/raster graphics editor, like Photoshop, Affinity Photo or GIMP, as it will be stuck at the size/resolution it was created at. If you try to upscale such a logo, it will get blurry and pixelated.
Even though Illustrator is the industry standard, you should be fine as long as you are able to work in the CMYK color space, which is used for print (more on this in a moment), and produce the right kinds of files, namely SVG, PDF, .EPS and PNG or TIFF.
Personally, I use Inkscape + Affinity Designer, rather than Illustrator. I sculpt my logos in the former, in black and white, and then transfer them to the latter to properly colorize them - using the CMYK color space - and export them to the aforementioed file formats:
Of course, you might want to learn how to use Illustrator, since it’s the industry standard, but as a freelancer you might also use other software.
If you want to learn how to use Inkscape and Designer, I highly recommend Nick Saporito:
COLOR MANAGEMENT
Logos are printed, for the most part, so it’s very important to colorize them using a color-accurate display, a color-managed app, such as Illustrator or Affinity Designer, and the CMYK color space.
Before you start making logos for clients, you should get a good display - preferably with a 90-100% coverage of the Adobe RGB color space) and a colorimeter. My personal unit is a 15,6" laptop 4K AMOLED display with a 95-96% coverage of Adobe’s gamut, calibrated with an X-Rite i1 Display Pro Plus (currently Calibrate ColorChecker Display Plus) and Calibrite CC Profiler (formerly X-Rite i1 Profiler).
Of course, there’s much more to screen-to-print matching than just a calibrated display, but without it you’re going to have absolutely no control over your colors.
You definitely should never colorize files for clients using a random, uncalibrated display.
If you want to learn about color management and display calibration, I highly recommend Art Suwasang of BenQ and Calibrite/X-Rite: