In a couple of weeks, I, a white woman, am going to explore the island
of Manhattan.   Are there any sections of the city that I should avoid
during the day?  Are there any sections which are generally ok during
a weekday, but not weekend (or the other way around)?  Are there any
sections that I should only visit when I’m in the company of friends? 
What sections are generally considered safe at night?   Which sections
should I be watchful for pickpocketers?   Do you have any other safety
tips that I should keep in mind?

Greetings Nmtmnh,
I went around the Net and found a lot of websites offering advice on
avoiding crime in New York City, and I suppose they answer most of
your questions. Some of the article whose links are listed below give
general tips on how to dress or what precautions to take when going
out in New York, and there are places to avoid mentioned. These are
especially mentioned in the forum-type pages, where people share what
they know and experienced. I hope you are able to read all of them:

Guide to Safety in New York City
http://www.nyctourist.com/nycinfo_safety.htm

VirtualTourist.com - Travel Forum / New York City
http://www.virtualtourist.com/f/p/a220/
- Quote from Poster: "Only places to avoid are some places like
Washington Heights and Central Harlem, but they're not too bad during
the day."

How To Avoid Being Mugged In New York
http://www.galactic-guide.com/articles/2R9.html
- This is quite a comprehensive guide.

Alain's Manhattan - with some tips on getting around Manhattan
http://cgm.cs.mcgill.ca/~alaint/ny.html

New York Stories - The LimoTour Inc.
http://www.limotours.com/tips.htm
- Note the "Personal Safety" section

About New York City! - Safety page
http://about-new-york-city.com/aboutsafety.htm

VirtualTourist.com - New York City Warnings Or Dangers Guide
http://www.virtualtourist.com/vt/cd624/8/?o=3&i=c

Red News - Rough Guide to New York City
http://www.rednews.co.uk/features/featsp10.htm

Manhattan ... A safe place to live
http://www.kstatecollegian.com:16080/issues/v099b/sp/n133/opn-Mastio-security.html


Google Search terms used:
new york city crime safety
new york city places avoid
manhattan places avoid
manhattan pickpockets avoid
manhattan crime safety

I hope this has been a most helpful answer. If you need anything else,
or have a problem with the answer, do please post a Request for
Clarification and I shall respond as soon as I can. Thank you.