I'm looking for a website for researching the best times of year to
travel (and times to avoid traveling) to certain destinations.

For example, I live in CityA and want to travel to CityB sometime, I
don't care when. But I want to know when are the high seasons (more
expensive & why) and when are the low seasons (least expensive & why).

There are many websites for countries that include this information,
often with a table for rainfall and temperatures, and there are travel
sites such as the following that provide some information:
http://www.fodors.com/

I am sure that we can help you with individual destinations, but one
or two websites that cover the globe in detail may be rare.