If you want to target the Welcome module only, you can do this if you make sure your selectors are more specific than the ones in the default CSS.
For example the CSS below will set only the background of the contained element below the sidebar with both the module and newuser classes.
Similarly the second selector matches a, p or h1 elements within that element, so should match all the text. You may need to add other selectors with comma delimiters if you have other text elements to match (h2, h3 etc.).
#sidebar .module.newuser {
background: #FFFF00;
}
#sidebar .module.newuser a, #sidebar .module.newuser h1, #sidebar .module.newuser p{
color: #FF0000;
}
Update: Note that this will also style the Question help section. If you want to only match the Welcome module, you can follow this answer, and add an inner div with an id (e.g. <div id="welcome">...</div>). Then set the CSS to match the nested elements of that div, e.g. #welcome a, #welcome h1, #welcome p { color: #FF0000; }