The first thing you have to decide is whether you are going to upgrade your domain to Windows 2003 or not. If you want or plan to make your new Windows 2003 server a DC, you must upgrade your domain. If you are planning to add the new Windows 2003 server to the domain as a member server, then you don't have to upgrade your domain. Upgrading the domain involves running Forestprep and Domainprep from the Windows 2003 CD on your existing server, then installing Windows 2003 to your new server and running dcpromo to promote it to a domain controller. If you're going to keep the old server as a DC and not upgrade the domain, then all you need to do is install the new server into the same domain as the old one, as a member server (i.e., DON'T run dcpromo).
Microsoft, by the way, recommends that you not run Exchange on a DC, although I've done it many times without a problem.
Here is an article that might be helpful:
http://support.microsoft.com/kb/822178/en-us
As far as actually installing Exchange 2003, the CD includes a pretty good wizard/checklist that walks you through the steps. I'm usually not a fan of such things, but I've used this one a number of times and it's actually pretty good. Once you've got the new Win2K3 server set up, whether as a DC or a member server, you just install Exchange 2003, configure it as you like, and then start moving your mailboxes.
A couple of tips:
1. Be sure to use the Exchange 2003 version of Exmerge to move the mailboxes. You must be careful, of course, to make sure that the users aren't logged onto their mailboxes while you are moving them.
2. Once the mailboxes are moved, the Outlook client should be able to find the new server automatically without any intervention by you or the users.
Hope this helps!