Edited: oops, it took me awhile to write this with phone interruptions, and the topic's already been covered.
So you're saying that you need more than 253 hosts in your network and need to change to a class A?
To re-IP a network is indeed a disruptive process so should be scheduled for off-hours. Obviously the number and type of equipment on your network will determine how long the process will take to settle out, and also how organized the network is to begin with, will affect the downtime.
In its simplest form, what you said is true, but I'll go into a little more detail.
Start with an inventory of all static equipment, i.e. routers and all managed switches, WAPs, servers, static printers, etc. Also look at your DHCP pool(s).
Next, plan the new network so you don't run into this again. Logically slice up the new network into several categories and map it out: routers / network equipment / static printers / servers / other static equipment / DHCP pools / "other" / etc.
Plan any routing changes
Now that you know where you're going, yes start with the servers and routers, particularly the DCs, DNS, and DHCP server(s), routers. The DHCP MMC doesn't handle DHCP server IP changes very well. Chances are you'll have to drop the DHCP server from any MMC you have it added to, re-add it, re-Authorize the DHCP server and de-activate the old scope. Make sure all of the servers are happy and are populating DNS with their new IP's -- though the servers don't tell you to, when I'm doing a re-IP, I like to reboot the servers and have them boot clean on the new IP. Inspect your AD DNS zone and remove any stale entries for the servers. Make sure all of the servers are happy and resolving each other.
Create the new DHCP scope and Re-create any reservations you have on the old scope with their new IP's but I wouldn't activate it yet.
Now that the core servers and routers are done and tested, make sure they can route to other networks and the internet, then work on other servers, Activate the new DHCP scope, and start working on managed switches, static printers, etc. and start testing workstations and user functionality.