Saving SVG files out of Illustrator is straight forward.
Simply press the save as button and select the SVG Type [ see image below ] to export the current file as an SVG.
An SVG options screen will pop up, providing advanced settings. In most cases you will not need to worry about the settings, Simply select "OK".
One exception, if your file contains live text ( text that has not been converted to paths ) in the Fonts section select the "Convert to outline"
The below settings [ "more options" selected ] show the most reliable export settings. If for any reason the defaults are not working, try the conservative settings shown below.