What he said, but I think you need to start smaller and less “business critical”. Start out by setting up a phone system at the “remote” school. It sounds like they’re tech-savvy over there, so they would be good people to have on your side.
There are lots of questions left unanswered in your post. For example, do you know how the T1 is terminated? Choices are a dedicated PBX system (maybe an old Panasonic or Avaya) or a “channel bank”. The first is a full-up “forklift” upgrade - everything including the wiring needs to be replaced. If it’s a channel bank, you can connect an Asterisk server through a T1 card and terminate all of the calls in the server.
The amount of traffic over the network for most “phone” applications is pretty lightweight, so the existing network will probably be fine.
One fine-print wrinkle is that your T1 may be providing your phone service AND your Internet connectivity. If that’s the case, you will need to coordinate with your upstream, provider.
There are a thousand ways to set this up so it works well. Here’s one:
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Set up an Asterisk server and extensions that talk to the Asterisk server. You can use almost any scheme you’d like. I like 4 digit numbers that start with ‘4’. Do this is a room somewhere so you can demonstrate the system before you shut down your old phones.
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Convert your phone numbers to a VOIP provider. Most of the big carriers support that now. If you school’s Internet service is coming in with your phone service, you can move the phone channels to support your Internet connection and increase your Internet Access by 25%. Your incoming phone service will now be delivered over the T1 as Internet Traffic.
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Set up your Asterisk server to route incoming calls to your front office/attendant/whatever. They can then route the calls through either attended or unattended transfers.
As you get further along, you can implement other routes.
The only part of this that I’ve ever found tricky is the FAX part. I usually end up going to an “off campus” FAX-To-Email gateway for my incoming FAX services. If you choose well, you can also get outbound EMail-to-FAX services. My VOIP provider provides a service like this. There are plenty of places around that can help you make this happen.