Which countries are the safest for the solo female traveler? What advice can you give to a beginner?

Obviously you should avoid countries with cultures where an unaccompanied woman is seen as fair game.

Other than that, be sensible, don't get drunk, don't go off into the countryside with strangers, don't flash your money or jewellery around, dress sensibly.

You should exercise caution and common sense, and review the other replies you got here, good advice.  I can only add that the occasions when I traveled alone, I found myself feeling lonely and unhappy about not sharing it with anyone else.

I am up and outside from time to time in Russia at all hours. I see women, teens (curfew at 10:00 pm, but some are a bit past 10:) men young and old out at all hours. Russia is safer than you would hear on news and embassy reports. The Caucasus is a different story. 

no country on this planet is really safe for a female alone ..best advice ..dont drink alcohol ..stay out of bars ..and the shadows ..trust no one ..and if you do have to go for a drink ..never take your eyes off it ..take it to the toilet with you ..i know two women who got their drinks drugged ..they cannot remember anything of the next 24 hours ..years later 

I have traveled alone to many countries, both for business and pleasure. I've never had any problems. I don't know what you think is going to happen to you- almost all the countries in the world are safer for women than most places in the U.S. The one rule- here and abroad- is not to get drunk in bars and go off with strange men. Tens of thousands of single women students go to other countries on study abroad programs every year- one of them, many many years ago, had a problem in Italy. That's not reason enough to fear traveling. Take precautions with possessions- some countries have skilled, pickpocket thieves, but the rates of violence against women and tourists is very low.