Learning about low-level graphics programming
The lowest level would be the graphics card's video RAM. When the computer first starts, the graphics card is typically set to the 80x25 character legacy mode.
You can write text with a BIOS provided interrupt at this point. You can also change the foreground and background color from a palette of 16 distinctive colors. You can use access ports/registers to change the display mode. At this point you could say, load a different font into the display memory and still use the 80x25 mode (OS installations usually do this) or you can go ahead and enable VGA/SVGA. It's quite complicated, that's what drivers are for.
Once the card's in the 'higher' mode you'd change what's on screen by accessing the memory mapped to the video card. It's stored horizontally pixel by pixel with some 'dirty regions' of pixels that aren't mapped to screen at the end of each line which you have to compensate for. But yeah, you could copy the pixels of an image in memory directly to the screen.
For things like DirectX, OpenGL. rather than write directly to the screen, commands are sent to the graphics card and it updates its screen automatically. Commands like "Hey you, draw this image I've loaded into the VRAM here, here and here" or "Draw these triangles with this transformation matrix..." take a fraction of the time compared to pixel by pixel . The CPU will thank you.
DirectX/OpenGL is a programmer friendly library for sending those commands to the card with all the supporting functions to help you get it done smoothly. A more direct approach would only be unproductive.
SDL is an abstraction layer so without bothering to read up on it I'd guess it would have different ways of working on each system. On one it might use semi-direct screen writing, another Direct3D, etc. Whatever's fastest as long as the code stays cross-platform..able.
The GDI/GDI+ and XWindow system. They're designed specifically to draw windows. Originally they drew using the pixel-by-pixel method (which was good enough because they'd only have to redraw when a button was pressed or a window moved, etc.) but now they use Direct3D/OpenGL for accelerated drawing (and special effects). Optimizations depend on the versions and implementations of these libraries.
So if you want the most power and speed, DirectX/openGL is the way to go. SDL is certainly useful for getting the most from a cross-platform environment and integrates with OpenGL anyway. The windowing system comes last but don't underestimate it. Especially with the stuff Microsoft's coming up with lately.
To be a good graphics and image processing programmer doesn't require this low level knowledge, but i do hate to be clueless about the insides of what i'm using. I see two ways to chase this - high-level down, or bottom-level up.
Top-down is a matter of following how the action traces from a high-level graphics operation such as to draw a circle, to the hardware. Get to know OpenGL well. Then the source to Mesa (free!) provides a peek at how OpenGL can be implemented in software. The source to Xorg would be next, first to see how the action goes from API calls through the client side to the X server. Finally you dive into a device driver that interfaces with hardware.
Bottom up: build your own graphics hardware. Think of ways it could connect to a computer - how to handle massive numbers of pixels through a few byte-size registers, how DMA would work. Write a device driver, and try designing a graphics library that might be useful for app programmers.
The bottom-up way is how i learned, years ago when it was a possibility with the slow 8-bit microprocessors. The direct experience with circuitry and hardware-software interfacing gave me a good appreciation of the difficult design decisions - e.g. to paint rectangles using clever hardware, in the device driver, or higher level. None of this is of practical everyday value, but provided a foundation of knowledge to understand newer technology.
Michael Abrash's Graphics Programming 'Black Book' is a great place to start. Plus you can download it for free!
If you really want to start at the bottom then drawing a line is the most basic operation. Computer graphics is simply about filling in pixels on a grid (screen), so you need to work out which pixels to fill in to get a line that goes from (x0,y0) to (x1,y1).
Check out Bresenham's algorithm to get a feel for what is involved.