If I am understanding you correctly, you've copied the esri.css file from the js.arcgis.com server to your local server. This won't work because that esri.css file references images and possibly other resources on the js.arcgis.com server. It references those relative to the location of the css file. The css file is on your server but those required images are not on your server so the popup and possibly other things will get "funky".
Your first inclination was correct. Edit the css classes in the main.css file. This will override the same css classes in esri.css. If your modifications do not appear to be working, first check that you've got the syntax correct and then be sure that the files are not cached by your browser.