I would definitely use LINQ to XML instead of the older XmlDocument based XML API. You can accomplish what you are looking to do using the following code. Notice, I changed the name of the element that I am trying to get the value of to 'loc' and 'lastmod', because this is what is in your sample XML ('name' and 'url' did not exist):
XElement element = XElement.Parse(XMLFILE);
IEnumerable list = element.Elements("sitemap");
foreach (XElement e in list)
{
String LOC= e.Element("loc").Value;
String LASTMOD = e.Element("lastmod").Value;
}