I know the below isn't the most efficient or advanced way to do thing - but I do think it is a way which almost anyone can do with minimal help and little downtime...
Assuming you're talking about a database for a CMS, you could you get the temporary URL for the new location of the website on the new server (we use external hosting, and they tend to set this up - the temporary URL is normally a suffixed ip address). Then go through the database replacing values referring to the old URL to the new, temporary one.
You can then restore the site on the new server (or whatever it is) also, replacing values for the new URL. This gives you the opportunity to test the database is functioning. When you are happy, you can just change the DNS (or A records or whatever), so the original URL points to the new server, change everything back (which should only take a minute or two with good find and replace procedures) and bobs your uncle. Its not the fastest way to make the transfer, but it should provide the least downtime to your site and peace of mind that it will all work.
What CMS are you using?