From the ones I've personally used, Autodesk Sketchbook is a nice drawing software. You can get some good detail in them taking the time but I usually used it for sketching purposes. It's totally free and really easy to use. It has different tools like a paint brush, pen, pencil, a sharpie like tool, etc. There is a pro version though, that includes a lot more stuff. More tools to work with and it even has an Animating feature.
Paint tool SAI is really good too. Though from what I can tell, the only free version they have is a 30 day trial. I think the full version is $70 US dollars. But it can do vector and raster based drawing/ painting, and really good with line art. There are A LOT of tools, I wasn't able to use all of them. For that one if you don't really know what everything is, I'd recommend a tutorial.
I hear good things about inkscape though I don't know that much about it. And I'm pretty sure it's free (Although I think that's vector based).
And I did a quick look at one other. It's called Krita and it's free as well. By the looks of it it functions pretty much like Photoshop. Though I think it excels with comics and manga (but I'm sure you can do more than just comics with it). A lot of brush variations. And a lot of tools. It's layout looks pretty much like a copy and pasted Photoshop layout too.
If you have a budget, Photoshop elements is (I think) $70 and a one time purchase, it's pretty much Photoshop with less tools. Although Adobe has a monthly plan for all of their software including Photoshop for $20 a month for students. If it's for non-students/ teachers, it's $20 for a single software, and it's $50 a month for all software. Better than getting the regular and paying full price and having to buy it again after an update.
Those are all of the ones I can list at the top of my head though. I'm sure there are a lot of other ones that I haven't heard of. You could try looking up speed drawing videos and seeing what they use too. If you're just beginning, I'd think that sketchbook would be a good choice to start with as it's a very self-explanatory software and you can do a lot with it, at least from what I've seen from others. There's also gimp but I never quite liked using that. Not to say that it's bad, but I didn't really like the UI personally. I started out with Paint.Net ^^ but I don't really recommend it.
Hope that helped