Within the next month or two, we're going to be getting brand-new computers at my office, and it's going to be my responsibility to set them up and network them. I've not messed with networking under Windows 7 all that much (or networking in generally, really), so I'm wondering this: Would it be possible to set all the computers up on the network so that all personal files and folders would be stored on a network drive? What would be the best plan of attack for setting up this network? We've only got about 15 machines total.
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Hmmm... 15 users. I don't think I'd want to manage 15 users and computers without AD. Creating a Windows Active Directory domain is going to afford you the quickest, most consistent avenue for managing users, computers, and resources. There are too many benefits to using AD to list here but my opinion is that implementing AD is the best approach to ensuring the long term stability and manageability of the environment.
As for your users files you can use Group Policy to configure Folder Redirection to redirect My Documents for every user to a network share, if you create an AD domain.
I wouldn't recommend using just a NAS. Going with Windows SBS 2011 Standard sounds like it would be about perfect. It will give you the centralized management you are going to want once you get things going. It can also be your file server and gives you access to a lot of other good stuff like Exchange, Sharepoint and Remote Web Workspace.
I don't know what your requirements are, other than centralized file storage, but it may be worth a look.
So as small as your shop sounds, here is what I would do: I would implement a server to be used as a common resource for all devices to access to store files. It doesn't need server class hardware, but it does need a server class O/S. I would use Windows 2008 server if that's what your familiar. To keep things simple, I would also not use active directory and just create matching user accounts on both the server and client. If you find your business growing, you can convert this machine to a domain controller and recreate the accounts and go down that road. The question is a bit ambiguous though. I can't cover every base in my answer here.
A small NAS like a QNAP or Synology might be an even better solution.
Friend of mine faced the same situation: 8 users with expected growth to 20 users max. in 2 years. He bought 2 Synology units (4 disks and 2 disks). Put 1 TB disks in the 4 unit and 1.5 disks in the 2 disk unit. The 4 disk unit (in raid 5, 3 GB effective capacity) is on-site serving several general use shares and each users personal home-dir. The home-dir is added to the "Libraries" in Win7. (one time manual action per user.) The 2-disk unit is setup in JBOD so it also has 3 GB capacity and is located 5 kilometers from the office in my friends home. The 4 disk unit backups every night to the 2 disk unit across the internet (rsync over ssh, so it's encrypted). We are in the Netherlands. Have have relatively fast and cheap internet over here so this is feasible. The office is on 100/100 fiber. The home is on 120/10 cable. Effective throughput office-home is about 60 Mb/s. (Different providers. Bottleneck is between them.)
Buying 2 NAS units + the operational cost for the internet pipes was cheaper than buying a single Windows server + maintenance contract + backup solution. If a disk breaks he can simply replace it. (In case this is on the 2-disk a full re-sync of the backup is needed of course.) If a NAS breaks he can simply buy a replacement and move the disks into it. If that happens to be the 4-disk primary unit the other one can temporarily be used as primary. (The share/user config is setup on both to make this easier.)