My two essentials:
A metric hex key (allen wrench) multi tool that goes up to six
A 15mm wrench.
This covers any fit adjustments and removal of the wheels if you need to put it in/on your car.
Since you're just starting fixed, chances are you're going to want to try different ratios. Get a chain whip and lockring tool so you can switch cogs.
If you don't put a cog and lockring on tight enough you will feel slippage on acceleration and when skidding or applying backpressure to the pedals heavily. If that happens follow this:
That concludes most of what I needed in the first month or so of riding. Everyone has different experiences but maybe ours will overlap a little.
Also - you mentioned you wanted lights. Beware of lights that require a button cell battery (most knogs or little lights). They usually take two batteries and replacements are sold in 3 packs for $11. If you don't want to pay the value of your light every time the battery dies get a rechargeable one. I have a Blackburn Flea which cost somewhere under $30 and it's small, usb rechargable, and brighter than my old knog light. Highly recommend if you don't need a full on pitch darkness headlight.