Low-level programming of modern video card in small handwritten OS

Low-level programming of modern video card in small handwritten OS

Consider small handwritten operating system in assembly/C++ from scratch
for 64-bit cpu loaded by GRUB as enviroment. The goal of OS is to execute
some small amount of code, required to render a 3D scene. That involves
simple disk management(filesystems and so on), memory management,
multitasking, module management (ELF). The point is how can you program
fast modern graphics card under described circumstances, like that ones
from NVidia or ATI?
What are the ways of:
setting videomode 1920x1080x32@60
rendering a pixel/line/polygon on screen
uploading uncompressed texture and displaying it on screen
uploading shader and running it
achieving V-Sync
What kind of tools, documentation, binaries, libraries, headers will be
required?