If you've registered a lengthy domain name (e.g. motherandtoddlergroupfor peopleformerlyconnectedwithhmsganges.org.uk) the only way to change it, per se, is to register a completely new domain name.
However if you simply want to have a shorter version of the name for, say, publication in a local newsletter (or on other websites), you can shorten it here:
http://tinyurl.com/
To get a random shortened address (such as tinyurl.com/fg4t7h), simply enter the over-long name into the blank field and hit 'Make TinyURL!'.
It might be better though to use the optional 'Custom alias' to choose something like tinyurl.com/gangeskids.
Once you've registered a domain name it should be automatically added to the list of URLs on the web; you don't need to do anything. Getting it picked up by Google (or other search engines) is another matter. They generally operate using 'web crawler' technology, meaning that they scour sites that they already know about looking for new links. So getting links to your site on other sites (such as that of local church or community groups) can be important.