Most bears (excluding grizzlies) will only bother you for one of a very few reasons:
(1) You come upon them by surprise. You can counter this threat by not moving too quietly--feel free to make noise as you travel, either by conversation, bells, etc. Traveling with a group helps here.
(2) You get between a mother and its cub. Don't do this! Ever! This is the most dangerous situation possible.
(3) They are hungry and you have food. Counter this by following the basic principles of the "Bearmuda Triangle" (google it).
If you are aware of these dangers, you are pretty safe from all non-grizzly bears. Nevertheless, if you do encounter a non-grizzly bear, follow the "look large" strategy, and yell at them. Do not run away.
Grizzlies follow no rules. They are completely unpredictable. You should of course follow the strategies above, but that's no guarantee of protection. Essentially, you just need to get lucky. You can do everything right and still get killed. Fortunately, there are very few grizzlies in the United States, and they are even rarer outside of Yellowstone National Park.
I wouldn't worry too much. The incidence of attacks by bears and lions is very low.