Select text between quotes?

You can use the Plugin BracketHighlighter to achieve what you want. First, you need to install this Plugin. Then, go to Preferences --> Key Bindings.

Add the following key bindings,

{
        "no_outside_adj": null,
        "keys": ["ctrl+alt+super+s"],
        "command": "bh_key",
        "args":
        {
            "lines" : true,
            "plugin":
            {
                "type": ["__all__"],
                "command": "bh_modules.bracketselect"
            }
        }
}

Then you can use the shortcut key Ctrl+Alt+Super+S to select texts between quotation marks.

There is a plugin specifically for this task: Expand Selection to Quotes. Simply put your cursor between two quotes and hit Ctrl' to select the text between the quotes. Hit it again and it will select the quote marks themselves.