Can one fly from Turkey to the Greek side of Cyprus?

no. Not sure why you would want to? if you are a EU passport holder or a Turkish cypriot you may freely move from Lefkoşa to Nicosia ( it is a divided city in the line of Berlin). If you are not one of the above you would have to fly from outside Turkey.

just want to go for a weekend and was told the Greek side was nicer... Don't know first hand... never been!
I hold a US passport and so does my friend...
what do you recommend?

Nice is relative, of course. I find the Turkish side nicer (more natural landscape, nicer people, better food). You could spend your weekend in Lefkosa and Girne, both very relaxed and original towns. If you want more posh and stressful, European style you should go to the Greek part of the city (Nicosia).

I just came back from a few days in Turkish Cyprus, and I completely agree with Utabe, even if what I actually saw from the Greek side is only South Nicosia, I much prefered Lefkosa ! And Girne, and Gazimagusa, and Salamis, and St Hilarion castle, and .... There's too much too see and visit and enjoy !!!! Go for it !!!

ok Turkish side it is... anywhere in particular you recommend for accommodations?

I actually chose our hotel according to some comments made on the forum by our helpful members ! And I was not at all disappointed ! We stayed at Hideaway Club in Girne (see http://www.hideawayclub.com/, tel.: 0090 392 822 2620), and the place is great, quiet little bungalows on the mountain side of Girne, with a beautiful view on the coast, with a very warm and friendly staff ! They can help you to rent a car if you want, which is quite useful when you wander around the country to discover its marvels !
Enjoy !