Now that you have your PCs (or Macs) and a server or NAS, you’ll want to join all this stuff together and get online, and this is where a network comes in. Once upon a time this would have been a complex task, but today it truly is plug’n’play (or in the case of wireless, just “play”).
When you sign up for Internet access with your ISP, you’ll probably have an option to upgrade from the standard bundled modem to a router at a discounted price. This is a pretty good option as most will have four (wired) network ports plus wireless. If you need something with more ports or more features then have a look at products from Netgear or Linksys.
The (first) question here is not, “What server do I get?”, but, “Do I need a server?”. Most people are just looking for some way of centrally storing files and the simplest way to do that these days is by using a NAS (Network Attached Storage) box. There are a few different options from companies like Netgear and QNAP supporting all kinds of features that you may (or may not) need. Check out the features you need and look for something that can be backed-up to an external drive.
Your other options (depending on your needs and resources), would be to look at Linux (I personally prefer Fedora over some of the new distributions like Ubantu) or Windows Small Business Server (for home offices you could even look at Windows Home Server)