PROGRAMSThere are several nice image editing programs for digital drawing or editing images. Some are freeware, some are affordable, and some professional programs are very expensive.
Here is a list of suggested art programs (prices where looked up winter 2018 and may have changed):
ADOBE PHOTOSHOPThe program I am doing most of my artwork and image editing with (I've been working with Adobe Photoshop for over 15 years now). It's a professional program with many options and features including an advanced layer system, processing effects, filters, adjustment layers, custom brushes, etc.
In the past Adobe's programs were available to buy as (expensive) full versions. They switched to a subscription system, so you can now get monthly subscriptions for one, several or the whole program range (Creative Cloud). As I am working with several programs, like Illustrator, Dreamweaver, Premiere, etc. I got myself the full subscription. If you only need Photoshop, it's already available for $10/12 EUR per month (Photography subscription).
It has an incredible amount of features, a lot of them customizable (like the tool bars, short keys, colors). Working with layer sets makes organizing your art a lot easier when working with a lot of layers. You can create your own brushes (or download brush sets, some are even for free). It would be too much to list all the features of Photoshop here (I recommend going to the Adobe website and reading more there, they also offer a lot of tutorials).
There is also a free app version Adobe Photoshop Express, but it's less a drawing program, but (what Photoshop actually is) - a program to edit your images.
CLIP STUDIO PAINTAfter working a while with Clip Studio Paint I can say, for sketching and coloring I even like this program a little better than Photoshop. It has most of the features that Photoshop offers (layers, effects, blending modes, adjustable brushes, etc.) but the sensitivity of the pen strokes (e.g. on the Wacom Mobile Studio Pro) is even better than Photoshop (and it seems to use less resources, so if you're using graphic tablet with touch screen, you can also easily rotate or zoom your canvas with your fingers). The coloring options are very nice, for example blending colors feels like drawing with real water colors. Files can be saved as PSD (Photoshop) files, so you can also easily work switching between the two programs and use the best out of each one.
PROCREATEWhen sketching or drawing on my iPad Pro, one of the best programs so far was ProCreate. It offers a lot of possibilities, including custom and adjustable brushes, layers, many features you know from Photoshop, and even a nice recording feature, giving you the chance to record the progress of your drawing and export a video file (e.g. for creating a time lapse). The files can also be saved in Photoshop format again.
OTHER PROGRAMSPrograms like Paint Tool SAI or Open Canvas are definitely nice and cheaper alternatives though. Especially SAI is very nice for sketching and inking, while Art Rage offers a lot of possibilities for imitating traditional media (like oil, pastel, etc.) in an authentic way, and Open Canvas for example is well known for its unique blending style (working perfectly together with a graphic tablet). SketchBook Pro is also very nice for sketching (I sometimes use it for digital sketching / instead of pencil) and color the picture in Photoshop later. It is specifically designed to work with pen tablets.
It's always a personal choice which works best for you, so best way is to try the different programs, as most of them offer a free trial period.
VECTOR PROGRAMSThe program I am using for vector graphics is Adobe Illustrator. It's not an image editing program intented (like Photoshop), but in first line for vector illustrations and designs (like often used for logos, decals, typographie, etc.). Like Photoshop, it's available as monthly subscription as single program or in the Creative Cloud subscription.
With Illustrator I do vector designs like decals, logos, layouts, or for example use it when I need to add a cut/contour outline to print files (for free form cutting).
Vector file formats are for example eps, svg or ai, pixel files would be jpg, gif, png, tiff, etc.