I was right where you are now about two months back. Some of the essentials I needed:
Allen keys, to make adjustments on the bike.
Helmet mirror, so I can see what is behind me without turning my head.
Extra tubes, 2 of them, just had a flat and using the new one.
Patch kit, to mend the tube with hole.
Lock, got a U lock and cable, Kryptonite, came with a pretty solid holder.
Plastic tire wrench, to change tubes and tires.
Front and rear lights, to enable me to bike in the evening/night.
Chain cleaner and chain lube. Self explanatory.
Of course a helmet, required by law.
Clothes pin, to pinch pant leg so as to prevent rubbing on chain.
Presta to shrader adapter, so I can pump up my tire with my home air compressor.
Tire pressure gauge, so I know what the compressor has pumped up to. I would suggest you just get a home full size pump with built in gauge.
Fenders, for when it rains. I got clip ons cause I dont like the look of fenders.
Chain breaker and quick link chain, for if and when the chain breaks, I can fix it.
Those are the basics that I needed to get going. I also got new pedal and handle bar to upgrade what I have. Next up is possible a saddle replacement and bar ends, and I think I will be almost all set.