This is a list of various features not supported in this and that browser.
Consult this list at once!
Also, remember that you need to enable WebDAV for some of the Office interoperability functions.
If "New Document" doesn't work: The ActiveX isn't loading as it should, you got a browser problem, and should consult the article above.
If "Save Document" (In Word or Excel 2010 or 2007) takes you to the local file system on a Windows 2008 client (terminal servers for instance): WebDAV probably hasn't been configured for the client, and you should consult the article below.
Enabling Desktop Experience for WebDAV on Win2k8
SharePoint has it's own implementation of WebDAV under the hood, even though it doesn't show in Server Manager. WebDAV is already configured and new document functionality comes out of the box for most installations and clients. It is on Windows 2008 clients that problems arise.
Also, as the linked article suggest, Windows Vista needs to have its Web Folders service started.