I'd skip both Grimaldi's and Lombardi's. Most NYers, myself included, will tell you they are tourist traps living off old reputations.
Personally I don't like DiFara, but it is a pizza mecca to many and you should make up your own mind and give it a whirl (and be sure you get get a square slice/pie). DiFara's Dom DeMarco is not going to be making pies forever. Be prepared to wait and watch for a bit. It is quite a trip from the bronx though. Even further, and my favorite NYC pie, is the original Totonno's in Coney Island. You could make a day of it and visit both plus get to see Coney Island.
Patsy's in Harlem is a good one too and that would be a lot closer to the bronx. It is the only coal fired place that is selling slices still. Might be some choices on Arthur Ave (Bronx Little Italy) but I am ashamed to say I have never been up there.
Be sure to skip off shoots of Patsy's, Totonno's, DiFara's in Manhattan. BUT Motorino in the EastVillage has the rep of being as good as the original one in Williamsburg Brooklyn, so that one is ok. You could go just within Manhattan and have plenty of great pizza. Check out slice.seriouseats.com for some good pizza maps of manhattan. Be sure to get at least one 'regular' slice at a corner pizza joint too