I'm assuming the bike itself has everything it needs (comfortable saddle, working gears, etc.)

I like to keep stuff to change a flat on my bike. This includes a hand pump along with a tube, patch kit, and a boot.
Lights are a given. With the time change, you'll start riding in dark more or when the sun sets.
I think reflective tape is overlooked. I think it's need just as much as lights are.

A rack is nice, but not needed. It helps keep sweat off my back and allows me to be more free while riding.
Of course a water bottle/thermos bottle holder.

Decent tires. If you're commuting to work or school you can't afford to be late. I like to get tires that have some flat protection so I won't get as many flats.

I also have an AirZounds horn on my commuter.

Lastly, a good lock. If this is a bike you're going to be riding five days a week, invest in a good U lock and possible a cable lock.

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For the bar essentials. All you need is stuff to change a flat. Lights too, unless you commute during the day.