Start here.

You'll need to be able to manage your DNS records and either own or be looking to buy a domain name. Once you have an account set up they walk you through setting the DNS records. MX records are the most important record to set. Some A records will also likely need to be configured (e.g. webmail.domain.com). And you should also set your SPF records, though that isn't mandatory, just a really good idea. At that point you'll basically pick a point in time where you cut over from your existing server to the new one. You change the MX records at that point and the mail starts flowing to the new system. In conjunction with that you also update the configuration of the mail client per the instructions that Google gives you.

That's pretty much it.