I'm in the Los Angeles area. The safest place for women to go traveling alone at night? Surely you jest! Los Angeles is not a safe place at all for women to be alone late at night. There are many safe community areas within Los Angeles county but few to none of those areas are really surrounded by the touristy type areas. With tourists, there is readily available spending money and that is also where you will find the professional homeless beggars and nut cases lurking around everywhere. Also, as for taking taxis, be prepare to spend a SMALL FORTUNE traveling that way. It will easily be in excess of $50 to go from the airport to a hotel near Farmer's Market. Expect a 3 mile ride to be at least $25 each way.
One of the things that I have learned from traveling around the world is that tourists rarely want to shop or go where the locals shop or go. Tourists want "Once In A Lifetime Experiences!". Locals want to avoid these places at all costs. I suggest you get a good book (maybe Insight Guide) on traveling to the Los Angeles area and start reading about the places that are available to visit, narrow down the areas you believe you would like to visit and then post some questions concerning specific areas. Los Angeles is an emormous area and anyone planning on getting around by taxi needs to have some good help from maps and a book to decide what area they want to visit and if that area can offer what they want to experience.

Here's a couple of suggestions:

3rd St Promenade in Santa Monica. Shopping, food and GREAT people watching. Of course the Santa Monica Pier is only a couple of blocks away.

Ronald Reagan Presidential libray in Simi Valley (about 30min away up the 405 to the 118)

Ventura Harbour - An hour away by car via the 101 or a more scenic route via PCH through Malibu.

Bob
Hating being on the east coast in these frigid temps!!

If you haven't been to LA since the 70's, the Farmers Market area is going to look SO different to you. They just built, in the last year or so, a huge shopping area attached to the market. So expect LOTS of crowds.

3rd Street is a good choice, as is Montana Ave ($$)in Santa Monica, or Melrose for some more shopping. If you will be here over a weekend, there are lots of flea markets in the area. Pasadena is a good city to check out- and if its the right Sunday of the month, you can hit the Rose Bowl flea market.

I know you said you had been to the Chinese Theater, but they just built the new Kodak Theater on Hollywood Blvd, along with a shopping center. Its still a nightmare on Hollywood BLVD, but for a quick touristy stop, it might be ok.

As for clubs, can't help you there too much, because what "was" hip last month is probably out by now:) But expect to pay around $10-$20 cover charge on the weekends for the good places.

Again, I would recommend against the taxi route. LA is just not a city designed for people without cars. You will have to call for a taxi every time you want one, and that will take a while to arrive, and then there is the cost. Traffic here is horrible, so the price for a taxi just goes up and up and up.