Google Apps are ready to receive mail for your domain even before you set the MX records. This is not written in the documentatuon, but is known fact.
When you change the MX recirds to Google's, there will be period of propagation of cource. During this perion, some emails will go to gGoogle Apps, some will go to your old mail server.
You can ask your users to check both Google Apps and old mail servers for a week,
or you can download the mails from your old POP3 to Google Apps. In this second scenario, assuming you have all the mails on old server, you can migrate all the mail to Google Apps as well.
Only "disadvantage" to this second download is you need to use "fake" email identification during the download - e.g. use something like [email protected] instead the real domain. When you try to set it up you will see what I mean.
Another disadvantage will be you need to do this as per user basis. If you have 4-5 users will be fast, but you might have 100 users or even more.
Finally during this download you can set up rules so the old emails will go to respective folders etc.
Hope this adds new useful info to existing posted answer.