My suggestions, for what they are worth:
Bring at LEAST one sandal and one closed toe shoe. If it rains you are not going to want to be walking around in sandals. If it is blistering hot, you are not going to be without sandals, at least part of the time.
I always bring more shoes than I think will be necessary. Yes, they take up luggage room. Yes, it's worth it. A change can be as good as a rest. (Keep in mind that I am a serious walker. When visiting NYC, I typically walk at least 8 miles per day.)
Go for the type of shoe that is comfortable to you -- preferably one you've already walked in a lot. You don't want nasty surprises.
For summer, I always bring one pair of casual sandals and one dressier sandal or shoe, plus at least one more closed shoe that I can walk in. Wedge sandals can work well (if you find a comfortable pair that meet your fashion requirements.)
If I were going to NYC tomorrow, I'd pack my Sofft Bernia gladiators, my Cons Hi-Ness, my Okalas (for going out to dinner or theatre -- I can walk in these for a couple of miles but they are not all day shoes). And I'd also bring my VC cage sandals, which despite their heel are surprisingly comfortable.
