Can someone tell me the best way to redirect to a new page so as not to anger the search engine spiders?
There are several ways, depending on what type of server you are on and what type of web page and whether you have the ability to add or edit an .htaccess file on your server.
I'm assuming you have just renamed or moved a single page, not your entire website?
Assuming further that you are still on the same server, the simplest way is probably to create a new page with the same file name as the old page like this:
<HTML>
<HEAD>
<META HTTP-EQUIV="REFRESH"
CONTENT="10; URL=http://www.yourdomain.com/newpage.html"><TITLE>This page has moved</TITLE>
</head>
<BODY>
<P>This Page Has Moved<BR>
Please update your bookmarks to
<A href="http://www.yourdomain.com/newpage.html">
http://www.yourdomain.com/newpage.html</A>
</p>
</body>
</html>
The part in red above redirects your visitors to the new site after 10 seconds. Don't set it to less than 10 or search engines may think you are trying to fool visitors by "stealth" redirecting.
Alternatively, if your host allows an .htaccess file, as Protoss implied you can use a 301 redirect:
Redirect 301 /oldpage.html http://www.yoursite.com/newpage.html or
Redirect permanent /oldpage.html http://www.yoursite.com/newpage.html
Or, if your page is PHP or ASP, you can add a script that will accomplish the same thing...