Hi there, Isc1027!
I'm also new to the world of map-making, but I have lots of experience working with digital artwork in GIMP, which is a free open source art suite, et al. There's lots of things you can do to clean up artwork that's hand drawn and scanned in like this. For example, this is your image after I opened it up in GIMP on my end, and applied a denoising filter, adjusted color levels and contrast, and also used a 25 hardness brush to get rid of a few of the more stubborn smudges:
PXL_20210611_004225657-01.jpeg
So you can definitely learn to do it too! Now having said that, you're asking what tools you should invest into learning. That's a tough question, because everyone has a bunch of different tools that they themselves use and recommend. I would say it's probably more important to pick a sufficiently sophisticated tool and then really learn to use that tool as well as you can. Having one powerful application that you know extremely well is often much better than having a bunch of different tools that you don't know as well.
Let's see here... so, I ended up choosing GIMP, and I got really good at it. But keep in mind that whatever is your "main" tool, well... the weaknesses of your main tool will become your weaknesses, lol. For example, with GIMP, vector art is doable, but GIMP's vector editing functionality is much worse compared to something like Inkscape. In my case, it worked out well enough because I prioritize general purpose artwork processing over vector art, but for something like maps, Inkscape likely has some features that GIMP can't compete with. Of course, that's not to say that you can't make amazing maps in GIMP/Krita, et al. It all just depends on what you want to be easy and what you want to be hard, I guess.
Sorry if this adds more confusion instead of answers, haha. It's tough giving specific recommendations for questions like this.