1.) Lights. At least one front and two rear blinkies. Extra batteries for the blinkies (almost all use AAA). A good "see stuff" headlight, which generally run at least $150, if you're riding away from streetlights.

2.) Lock. Up to you how much security you want.

3.) Something to carry cargo.

4.) Stuff to fix a flat. A tube or two, tire levers, a patch kit, some tire boots, and a pump (I recommend a Topeak Road Morph G or Tubro). Also practice taking your tire off the rim at least once at home.

5.) Fenders if you plan on riding in the rain or on wet streets.

6.) Multitool.

7.) Gloves.

You don't need bike shorts or a jersey for riding around a city.

You don't need a bottle cage or water bottle (just toss any old water bottle in your bag/panniers).

You don't need a computer.