In order of priority (bike bottle/cage are obvious, so I won't cover them):

-Helmet
-Pump, Patches, spare tube, Tire levers, Multi-tool all in a seat bag
-I will second checking your rim strips. If they are rubber replace with Velox, anything else if they look OK, give them a shot, if you have issues, replace with Velox. All but my Sirrus (factory wheels) have Velox rim tape, never a single issue.
-Strong legs

That will get you rolling for now. Down the road (so to speak) you might find you want gloves, bike jersey/athletic shirt, bike shorts, Bike computer, and all kinds of other things. But the basics are listed above. I don't even have a multi-tool, and I need one. My canal path bike has allen wrenches in the seat bag, so I am covered there.

Enjoy the bike! (oh, and Podium bottles are wonderful, they make them insulated too...)