|Client Licences - that's just the way it works. With Windows Server you need a CAL for each computer that connects to the server.|
You talk about "logging in" to your server, but I don't think you are clear as to what you expect from it. If you just want to access files on a shared drive then you just map a network drive. You would very much want to use a VPN to do this from the outside world; otherwise your files are potentially open to every Tom, Dick, and Harry.
I'm getting the impression that you don't really know what the purpose of a server is. It's not good to make business decisions that are going to cost thousands of dollars without being clear upfront what you want and how it is going to affect your business.
You don't run programs off servers. (That's not entirely true. There's a thing called Terminal Services that lets you do that, which can be useful for large organisations, but it would be totally inappropriate for you. For starters you need another set of client licences for TS; for seconds you don't save money on software because you still have to buy as many copies - or licences - as there are people using the software.)
Servers are used to centralise resources and to make management of client PCs easier. So you share files and printers using a server, you might run a database server, a web server, a mail server, a calendar server,... All sorts of things. But none of these seem to be applicable in your case - other than perhaps sharing files and printers; and there are much cheaper and easier ways of doing that.
Forget about the network infrastructure - you would'nt need to change any of that except adding a switch to give more capacity.
I think you are going about this from completely the wrong direction; worrying about implementation details when you don't know what you want to implement. You need to make a proper business case for this detailing your requirements and the costs of implementing those requirements; otherwise you are likely to spend a lot of money on something that is of very little use to you. Include in those costs training for whoever is going to manage your network.
Rather than taking advice from Internet forums, from people who don't know your business and what you are trying to achieve, I would recommend that you get some advice from a consultant; it'll cost you a few dollars but it could save you thousands.