Are you actually looking for an education, or do you just want to get the job done?
Migrating IP address space is always an "interesting" endeavor. I've seen it accomplished many ways, so more painful than others.
The most straight forward way that I have seen is to start by rehosting the web and mail server to the new systems, so that you know that they are working. Moving the web server can be difficult if there is a lot of dynamic information, since both systems need to point to the dynamic content. If there is a DB server involved, check into replicating that content between the two systems.
Once you know that the web and mail servers are working, you change the DNS entries to point to the new servers. In the case of mail, make new MX record entries with a lower number for the newer server. After several days, you should see requests to the old servers drop to nearly 0. If not, you've forgotten something. DNS, unfortunately, is not instantaneous. There are many caching DNS servers on the net and it takes quite a while for changes to propgate to all systems.
Once all of the traffic is hitting the new servers, you can decommission the old ones at your leisure.
If they would rather just move the old servers to your existing IP addressing scheme, you can install secondary IP addresses for those machines so that they will answer requests from both networks, and then decomission the old IPs once the traffic has migrated.
Moving the desktops is even easier and does not need to be coordinated with the servers at all. Since most networks are DHCP, it is simply a matter of changing the DHCP configuration file.
If the old network is not DHCP, I would suggest installing a DHCP server and migrating desktops one at a time to the DHCP server. It is a trivial change to the desktop, but you must visit each one, unless you have something like Tivoli to roll out configuration changes.
pansophic