Switching to 16-bit mode is, I would say, a last resort. I would only do that if you had no other choice.
If you are in 32 or 64 bit mode, drawing pixels to the screen is fairly easy.
What is not easy in 32/64 bit mode is switching from text mode to graphics mode. If you can do that with GRUB or in your custom 16-bit boot loader, that would be the recommended approach.
So, switch to your desired graphics mode when booting, before you switch to 32-bit/64-bit mode, and do everything else without going back to 16-bit mode, if at all possible.
HDMI is handled automatically by the video card. You shouldn't have to do anything special to get HDMI to work.
Once you are in 32/64-bit mode, you simply find the address of the video card frame buffer, and start changing memory address values, and the pixels will appear on the screen.
Details are on the wiki page here:
Drawing In Protected ModeLet us know if you have any questions or run into any problems.
Good luck.