As a New Yorker, I'd say - wander!
Given that you like arts, culture, and eating, New York has so much to offer - you won't get bored in 10 days. And barely there is any better way to discover the Big Apple then wandering around. Yet, make sure you wander in such boroughs as Soho, West/East Village, Chelsea - Downtown in general. Not because other boroughs are unsafe, these are just the beautiful ones and with nice restaurants.
I wouldn't say "Avoid Times Square", yet don't have high expectations. Definitely walk in Central Part - by the time of your trip it should be blooming.
Explore Brooklyn downtown- the part located closer to Manhattan. Brooklyn is huge, yet I wouldn't say the entire city is worth checking out. Focus on Brooklyn Heights. Watch the sunset on a bench at Brooklyn Promenade. There is also a magnificent view from the Dumbo Park - or the park of two bridges. From there you can literally walk Brooklyn Bridge and find yourself in Manhattan - great experience.
And finally, when choosing the tower to visit for a skyline - I'd recommend the Freedom Tower, as it has the best view of the city.
I'm sure you're aware of the greatest museums in New York, just a little tip: MoMa is free on Fridays 4pm-8pm and Metropolitan museum ticket price is up to you, because it is donation-based system. Normally New Yorkers pay $5-$12 per person there.
Enjoy your staying, I hope it will be memorable!