Well for Cross-Browser computability - I almost always use a css reset.
Something that may be of interest to you is Normalize - http://necolas.github.com/normalize.css/
which helps things looking / acting the same cross browser.
For IE you almost always have to do conditional stylesheets for versions lower than IE9 - like the PNG fix for IE6, various padding / margin issues in IE7 and IE8 -
so for that take a look at
Possibly look into a css framework that is already "cross-browser" friendly. Bootcamp, Foundation, Skeleton, etc etc...
Other than that, all I can say is read up on some proper techniques in some blogs like "CSS tricks" and keep on learning.
Hope this helps