Like all 8-bit home computer vendors, they made their own audio and graphics ASIC chips. Sound programming consisted of variants of ADSR (Attack Decay Sustain Release) programming which was replaced with industry standard MIDI with the Atari ST. Graphics programming was based on different text and pixel video modes and player missile graphics/sprite programming. Text modes allowed characters of single/double/half character widths, and double/regular heights. Pixel modes were between 1, 2 and four bits per pixel. The added boost was with player-missile graphics which was an extension of the 2600 console and sprite programming (similar to the Commodore 64 and TI-99/4A). You programmed in little pixelmap images then set registers to position them. Add some extra code to automatically update their positions and you had animation with a very simply physics engine (velocity, acceleration, collision detection). Some video modes supported 16-colors (rainbow or various shades of a particular color). Some programmers developed an APAC mode (any pixel any color) where two video frames are used to pick two different standard colors for each pixel to make a new pixel color. - effectively 24-bit color).
De Re Atari was considered the holy book on programming.
They had their own trackball, light pen and graphics tablet controllers, but didn't include an RS232 interface - that came as an extra podule that plugged in through the serial bus. Other home computer systems had these as standard, so you could use generic dot-matrix printers, modems and frame grabbers.
This was a brief period of time in the industry where international standards/methods were too expensive silicon wise, so everyone tried clever hardware design techniques for IO, graphics and sound.