Simply ensure that, while your old email account is still active, that you divert all incoming mail to your Gmail account:
http://email.about.com/od/gmailtips/qt/et_collect_pop.htm
Then send your replies via Gmail, asking people to note your new email address.
Also, of course, send out a 'change of address' email to everyone you know. For most purposes it would be best to do that from your Gmail account (so that people can simply click the option to add the address of the incoming mail to their address book). However some organisations might prefer you to use your old address to send the message, so that they can see that the information really has been supplied by you. (If in doubt, send the message via both addresses).