Saving SVG files out of Illustrator is straight-forward.
Simply select 'Save As' in the File menu and select the SVG type (see image below) to export the file as an SVG.
An SVG options screen will appear, providing Advanced Settings. In most cases, you will not need to worry about the settings. Simply select "OK"
Note: If your file contains live text (text that has not been converted to paths), you must select the "Convert to outline" option in Fonts > Type
The save settings shown below provide the most reliable export settings. If for any reason the defaults are not working, you can try the conservative settings shown below.