It depends.
If you have a public IP address that you are forwarding to your server, you simply need to change your router to forward to the new server address. I realize the address you noted in your post is not real, but if it is in the 10.x.x.x block then this is most likely the case.
If your server is assigned a real public IP address and you will be getting (or already have) another public IP for the new server, you will need to have the company that is providing your DNS servers (most likely your ISP) update your MX record. It can take up to 24 hours for the new information to propogate, but I think you will find that most DNS servers will pick up the change within a few hours. There will be a period of time where some DNS servers report the new address and some report the old.
Assuming you are using public IPs, there are a few ways to deal with the situation (in reverse order of preference).
1) You can have multiple MX records with different priority. The MX record with the lowest priority gets hit first, if it is unreachable then the next until all servers have been tried. You could have a MX record added for your new server with a higher priority number (while leaving the existing in place). Wait a day, then disconnect your old server. At that point, you would have the primary MX deleted (or leave it and change priority if you would like to keep the old server functioning as a backup) and change the priority on the new record to 0, making it primary. Using this method, your incoming mail will be delayed during the transition, but it will go to the correct server.
2) Have your MX record changed to reflect the new address and configure the old server (or a router) to forward all traffic on port 25 to the new server. This should be fairly simple and should not cause any delay in inbound mail delivery, so it is probably the preferred.
3) Change the IP address on the old server to something different and assign the old address to the new server. If you are not running any other public services on this server, then this is by far the easiest solution as you do not need to worry about the MX records.
MX records are not difficult, but your experience will depend upon your ISP. Some of them can really complicate the situation, so be sure to allow a few days for them to make the changes in addition to the time it will take to propogate.