There are really few sports that require such little gear as running. The beauty of it is that one can get started with any old pair of sneakers.
Personally I bought nothing for the first couple of months. I feared that if the novelty of running wore off after a few weeks, I would have purchased new gear in vain.
Obviously being a woman a sports bra offering proper support is a must, so I think this is an essential item too.
And lastly, when running in the dark winter months I find that a vest with reflective material is quite practical for being visible, along with winter tights and a wind jacket.