You definetely want a heatsink/fan on the processor.
Nothing fancy though. Buy a retail box version of the PIII 1GHz - it comes with heatsink/fan. If you buy an OEM chip, i.e. one that comes in a static bag, then get a heatsink/fan rated for that chip.
Case fans install in the computer case/chassis. Processor fans install on the processor. Whether you need a case fan depends on your climate and how much other heat generating (hard drives, video cards etc.) devices you have in the box.