This is great advice.... If this place is anything like a couple of them I've seen before though? They likely decided to become primarily a "Linux shop" in the first place because they were unwilling to spend much on I.T. -- and somewhere along the line, staff deployed Linux as a way to keep old/obsolete hardware functional.
Assuming you can get some kind of workable I.T. budget in place, I think you want to start by analyzing what's exactly going on, on the server-side of things. Windows Server 2003 still in use? Where and why? Is there an Active Directory master keeping all of the user account logins? How many servers are just doing basic file/print or web services for various things?
In the last 2 jobs I've had, it made sense to invest in a relatively high-spec server to run VMWare ESXi and create virtual servers in place of the older, physical systems. Right off the bat, you get a cost savings in electrical power usage (less heat generated by a bunch of older servers in a computer room, etc.). If they have "legacy" apps that would be problematic to get running properly on a current OS, at least you can virtualize that old environment and run it on the new system where making regular snapshot images of the whole thing is trivial. And you often remove physical constraints on the maximum available storage space too. (Old servers with SCSI RAID cards may not support drive partitions over a certain size, and you may not be able to add hard drives of the capacities you typically see today.)
On the PC workstation side of things? Anything running XP should be budgeted for complete replacement, IMO. Yes, some of those systems can easily run Windows 7 -- but by the time you buy the licenses for them, you're probably spending about as much as the used hardware is worth in resale value, if not more. Exceptions might be any laptops bought in the Win 7 era that just had XP loaded on them because that was what they preferred.... On those, maybe you can just load a Win 7 recovery/restore disc that came with it to begin with and get it current at no cost except for your time.