A few mentioned avoiding downtown. I agree, it sucks. But not because of dangers of traffic, etc. But because it isn't any fun.
I live downtown and learned downtown. Started riding at night when I was new so there were fewer "obstacles" to look out for.
After the first season of this, I just started riding to work everyday (also downtown).
I have to say, I would rather ride downtown in full weekday traffic than ride in the burbs. The soccer moms and weekend drivers are way more dangerous.
Having said that, the key is to always be on the lookout for potential dangers. I think downtown riding trains you the best. Besides cars, you got pedestrians stepping off the curb, street car tracks, street cars dropping people off in front of you, bicycles coming out of no where, random animals (skunks!), trains, parked cars, narrow streets, huge pot holes, construction vehicles, street sweepers, etc.
Learn to avoid all those and keep that training and you will be comfortable riding anywhere... Even the burbs... Lol