Introduction
Graphics cards are a very complex topic, I'm confident in saying that they are the most complex subsystem you'll find on a PC.
If you ever found yourself lost programming an XHCI (USB 3.0) or an old RTL8239A network interface card then be prepared because this is much more complex.
Graphics controllers are the products of a very competitive marketing - rarely a vendor opens the specifications and when it does, it gives an intentionally poor support.
If you add that the hardware itself deals with: codecs, audio (yes, audio streams too), 3D programmable pipelines, video signals and video outputs, surface formats, media formats, DMA and memory remapping then you can see that it is not an easy task to program a video card.
The better approach, in my opinion, is to "retrace the history" of the video cards.
Start from the MDA then move to CGA then EGA and finally to VGA.
The VGA legacy is still supported, the specifications can be found here or in the first part of this PDF from Intel.
You can program the VGA without the BIOS "easily" - meaning that it is an already well-known and documented hardware architecture (but not necessarily easy to configure).
I don't remember if the previous adapters were subsets of the VGA or not, if not they aren't supported anymore probably.
You can try with a virtual machine or an emulator.
When you are satisfied with the VGA you can move to the SVGA.
Here come the troubles: as Wikipedia confirms, the VGA was the last truly standardised video card/adapter interface:
Unlike VGA—a purely IBM-defined standard—Super VGA was never formally defined.
The organisation VESA standardised a BIOS API called Video BIOS Extensions to allow the use of SVGA cards to driverless OSes but that's not what you were looking for.
You can try reverse engineering a VBE BIOS but I think it will be a nightmare - a senseless stream of writes to IO ports and MMIOs.
Making sense of tenths of configuration registers without any reference is almost impossible.
Note that we are still talking about 1998 technology up to this point.
After the VESA VBE effort, no more standard interfaces have been published - the only reliable way to program a video card with less than 20 years is by signing an NDA with its vendor.
Luckily, recently (actually, not anymore), Intel entered the market with its Intel GFX (a.k.a. Intel HD Graphics) cards.
Intel never aimed to manufacture top-of-the-notch video cards, not even closely - so they can be open about their architecture since that's not their core business.
The result is this marvellous set of Programming Reference Manuals that describe the functionality of their video cards.
Complete with (traditionalistic) minimal information to program them.
In general, hobbyists stops before this point (at the SVGA checkpoint), because the hardware has become very complex and the efforts very huge.
For example, my Haswell integrated video card is documented with 17 PDFs of about 250 pages each (on average).
The display part is documented in a PDF on its own, the framebuffer has disappeared in favour of Display surface and the display part alone of the hardware is this:
While this may not be very comprehensible, it should suffice to get an idea of the numerous technology that a programmer must understand before programming a modern video card.
You can surely take a look at the Linux source code but beware that the Linux kernel is no usually of immediate understanding even for simple controllers - it is not a toy OS, it is a real OS with its own API and interface that must fit the hardware interface (actually the other way around).
Furthermore, only the Intel and AMD video drivers are really open source, the others are either proprietary or just a bunch of undocumented code.
Brief outline of common VGA modes programming
If you just want to program the VGA (a very respectable task indeed!) you can start by setting the video modes 03h (text mode) or 13h (graphics mode).
Video mode 03h
The frame buffer is at 0b8000h (physical address), usually accessed as 0b800h:0000h as it is handy to have a zero offset.
The screen is made up of 80x25 characters, each characters occupy a word (16-bit) in the frame buffer.
The low byte is the character code - the character map used will associate a glyph to a code (e.g. 41h to A).
The high order byte is the attribute byte - the low nibble is the foreground colour, the high nibble is the background colour.
More information can be found in the EGA/CGA/VGA links above.
Video mode 13h
It is a graphical mode with 320x200 pixel, the frame buffer is at 0a0000h (physical address) usually accessed as 0a000h:0000h for the same reason of above.
Each pixel is a single byte, the value of the byte selects the colour of the pixel.
The default palette can be changed by programming the DAC registers (3c7h, 3c8h, 3c9h for the VGA adapter).
Answers
A graphics card has two modes: a text mode, and a graphics mode.
Not necessarily, today this distinction may not exist anymore.
The MDA had only a text mode.
EGA, CGA and VGA and SVGA had both.
The modern approach is to draw the text, however during boot or during particular situations (e.g. BSOD) a basic video driver in text mode is used.
This driver probably uses a BIOS service since the video driver may not be available/reliable.
You can write data to a graphics cards using BIOS
Up to the SVGA era, then BIOS support was discontinued.
How do I know what is the base address of the video memory of the graphics card, is this done by probing the PCI bus to get the base address, or is the base address fixed (just like the COM ports base addresses is fixed for example)?
Video cards have been connected through the history to the ISA, PCI, AGP and PCIe buses.
Only the ISA bus wasn't configurable (at least not from the beginning), the others had configurable BARs (Base Address Registers) per function (the smallest addressable entity in the PCI bus).
In order to get the base address of the MMIO registers of a video card the PCI or PCIe bus must be enumerated and the standard registers in the configuration space must be read/set.
Dealing with PCIe is not as easy as dealing with PCI.
Note that not even the UARTs have a fixed address, they are configured by default to map to the legacy (3f8h, 2f8h, 3e8h and 2e8h) addresses but the hardware was (is?) in a SuperIO chip behind a PCI-to-LPC bridge that emulated a PCI-to-ISA bridge.
With the advent of the Intel platform hub architecture (i.e. the death of the north and south bridge) the SuperIO chip eventually made it into the PCH or moved behind the SPI controller.
Are all graphics cards accessed in the same way, or do I have to create device drivers for all available graphics cards?
Each graphic card is a beautiful vicious creature on its own.
A device driver is needed for each model.
Some driver can be reused for a whole family of models but this is not true in general.
