How do I know if there is a pest problem? Do I avoid certain areas or is it inevitable?

I think a good tool to use when apartment hunting is rentenna.com. This site gives you a brief summary of the building's reported issues (noise, pests and otherwise). I would recommend drawing comparisons between buildings of different areas to see what areas trend lower with regard to pest issues, and of course, take the information with a grain of salt.

its new york. even expensive manhattan penthouses have roaches, and 5th ave was shut down with bedbugs not that long ago. if you dont want to deal with bugs youre going to have a hard time here.