Sublime Text has Vi emulation (implemented in the Vintage package), but it's disabled by default. Go to Preferences | Global Settings - User in the top menu and make sure you have this in there:
Now you should be able to use many Vi commands.
If you don't want to use Vintage, you can press Ctrl+Shift+Space and the selection will expand to the current scope. Strings normally have their own scope, so the whole string should be selected (probably including the quotation characters themselves).