I'm assuming that your Internet site collection and intranet site collection are not the same site collection with what I'm about to write. I am assuming, however, that they are housed in the same web application. If that's the case (and I understand enough of the specifics), here's how you'd carry out what you're trying to do:
Establish a Web application to house your site collections. You've already taken care of this (since you have your site collections available to you internally). In setting up a Web application, it (by default) is exposed at a URL (or server:port) through the Default zone mapping. For our purposes here, we'll assume that this is the URL through which you want to access the site internally (on your Intranet).
In order to expose your site collections via the Internet, you're going to want to extend the Web Application housing them. This is done through Central Admininstration > Application Management > Create or Extend Web Application. In extending the Web Application, you're creating another IIS site with (ideally) a publicly-accessible URL that can be exposed to the Internet. You'll be asked to pick a zone as part of the process; given your needs, I'd go with "Internet."
At this point, the Internet zone (you just extended) is still setup to use Windows authentication and Active Directory as it's membership provider. Though you probably want to keep AD as a membership provider (based on what you've stated), you'll probably want to look at enabling Forms-Based Authentication (FBA) on your Internet zone. Microsoft has a video on that here: http://technet.microsoft.com/en-us/windowsserver/sharepoint/dd355701.aspx. Note: you won't want to use the SQL membership provider if you intend to continue using Active Directory accounts. Instead, you'll have to wire-in the Active Directory Membership Provider for FBA. Some info on that can be found here: http://blogs.msdn.com/solutions/archive/2007/08/27/forms-based-authentication-fba-in-wss-3-0-moss-2007.aspx.
At this point, your Default zone site should use NTLM and an intranet-available URL. Your Internet zone site should use FBA and have an Internet-available URL. You'll need to enable anonymous access on your public site collection for the Internet zone. This is done through a combination of Central Administration changes and changes from within the site collection itself (http://www.mindsharpblogs.com/ben/archive/2007/02/11/1557.aspx). Important point: when going into the site collection to enable anonymous access, be sure to go through the Internet URL; don't go through the default zone (i.e., the intranet zone).
With all of these things in-place and your site collections (or more specifically, the IIS site servicing the Internet zone Web application) wired-up to the outside world, you should be good to go.
I made a number of assumptions as I wrote this, so you may (obviously) need to adjust. Setting up anonymous access isn't overly hard, but there are a lot of steps to it. If you hit hiccups along the way, don't be afraid to search for answers. Many folks have done it successfully ... but more often than not, there are challenges along the way.
Good luck!