So for the splash screens all I do is take an image, then convert it down to only a few colors, sometimes 4, other times more, depends on how it looks. If its more than 4, I then need to choose which colors get changed to which, and sometimes I may divide them in half, so a color might be light green on a head but dark green on a body, or whatever. And then I need to make adjustments here are there to make it look better, working on a pixel at a time. It can take some time, but I enjoy it.
As for the backgrounds, I tend to take inspiration from places and run with it. For example, the Moscow background is literally the splash screen from Tetris on Game Boy, then I just expanded it a bit keeping the same style and throwing in a few more of Stalin's palaces. Havana came from a picture of Havana, I saw the same patterns on every building, just in different heights and colors, so I simply recreated that pattern and then adjusted colors and heights of the buildings. As for Honduras, I worked with actual pictures of palm trees and Mayan pyramids.
With splash screens and backgrounds you also have the issue of tile limits, and I used the DMG limits, not GBC as I intended for it to be GB game. So you gotta be smart and cut and paste tiles where it won't be noticeable to get it within the limits. Ultimately I had to make it a GBC game because it just ran too slowly on the DMG that it wasn't fun.
The sprites, also pictures, cartoon pictures, or even sprite pictures that I then ran thru the same processes. Since they are on a small scale and only have three colors you really gotta put some effort into it to make them look good, but since they're small it doesn't take all that long.
I actually studies architecture so took a lot of classes on using phot editting and 3D modeling tools in grad school, and all of that translates really well into making Game Boy art. Well maybe not the 3D modeling.
Hope this helps!