Well, humans are hominids, and hominids are primates. Whether hominids are apes or apes are monkeys is a matter of semi-arbitrary distinction. I mean, one could argue that mammals are reptiles and all vertebrates are fish, and you'd not technically be wrong in the evolutionary sense.
(This is also why I'm leery of the claim that birds are dinosaurs. If you're going there, it might be more accurate to claim that birds are saurischians, and saurischians and ornithischians are archosaurs which are semi-arbitrarily lumped together as "dinosaurs".)
But I'm fine with the argument that humans are apes and/or monkeys, if for no reason other than to troll the fundies.