You should be able to use a 555 in combination with an L297 and an L298 to drive small (floppy-drive-sized and a bit larger) stepper motors.
While Olin is right about acceleration in medium to high performance cases, for comparably slow speeds and light loads you can just start issuing step pulses with the 555. Microcontrollor sequencer solutions do have a lot of advantages, in cost, board space, movement profiling, and flexibility however some of the IC circuits can be advantageous for their ability to modularize potentially tricky parts of the problem such as PWM (chopping) current regulation - something that can be done in software, but may add a lot of complication to a learning project.
Topics like the pulse sequence for a stepper motor, internal workings of the 555, etc are covered in numerous references which any search engine will locate. The question & answers format is better for the problems that remain after doing some research.