1) Re-validate the HTML and CSS. In my case, I find that in 30-50% of the cases, something I recently changed has caused my markup or CSS to no longer validate. For me it’s usually a typo (maybe a semi-colon when I meant it to be a colon, or something equally stupid and hard to spot).
2) If it’s just a minor cosmetic issue, I’ve finally come to believe that it’s often OK for a site to look slightly different in different browsers.
3) If it’s a more serious issue (like your positioning problems), I’ll do a Google search. Maybe it’s a known issue with some workarounds. But if that doesn’t get me anywhere, I’ll probably ask for help here on css-tricks.