I am getting CAT6 strung around the house to set up my first network.
Excellent. Don't forget to have good N-based wireless as well.
1. Can you recommend a good gigabit switch with at least 12 ports on it?
The suggestions already made are excellent choices, you can't really go wrong with gigabit switches these days. You will, however want to verify the total backbone bandwidth if you plan on having busy switches. Because a lot of switch say they're duplex, meaning all nodes can send and receive at the same time, this is largely true but to save cost some switch manufacturers actually cut down on all the interconnected traces. So a switch with 16 ports in full duplex should be able to have a total backplane/backbone bandwidth of 3200gbps...most only have half that.
And yes, it makes it a huge deal if you have only one switch to service all the nodes in your home.
2. What would you use for a server? Right now I don't plan on streaming scads of movies or anything. I am looking for something that is cheap, quiet, small and easy to purchase or easy to build.
Disk I/O will be a huge limiting factor. Consider running multiple drives, plenty of ram, and a multi-core CPU. Run those multi-drives in a performance RAID configuration like RAID 5 or better yet RAID 10. OR consider splitting your servers in two depending on what you are doing:
Server 1 - SMB/SAMBA file server, web server...etc. <--- CPU is less important, moderate amount of ram, phenomenal disk I/O
Disk I/O is essential here and is your sole concern when putting together a file server for use at home. If you decide to enable SSL for HTTPS on your web server, then clock rate on the CPU is very important.
Server 2 - Windows 7 media center. <--- multi-core CPU with fast processor, with lots of ram, decent disk I/O, fast network
The Windows 7 media can be spawned in multiple processes, and will also do transcoding if you have the appropriate transcoder plug-in installed. Multi-core is the most important with a fast CPU. Disk I/O is very important but not as much as network I/O either. Gigabit is fine, especially if you have a switch that can do Quality of Service on video and multi-media.
3. What OS should I use for the server? I have literally no Linux experience. I just want something simple and secure that I can set and forget.
Linux for file and web server. No better time to learn than now.
Windows 7 Premium, Pro, or Ultimate so you get Windows Media Center. With a media center extender integration is seamless.
Most *nix based multi-media centers are capable, but not as painless as Media Center.
4. Do I even have to have a server?
No. But it beats carrying everything on a laptop or having a desktop that your little brother can steal.
5. What software is necessary for a server besides antivirus? Do I need firewall software, or is that built into the switch?
No. I'm even hesitant to run antivirus on servers...you do not get virus' by watching movies, or storing your personal information. You could get them if there is a digital rights management exploit, but anti-virus on a server will not catch that. Run anti-virus on your desktops, laptops, and clients because you generally get virus' because people's internet browsing and email habits are shit.
If you do end up running an email server, you will want anti-virus to monitor the server's transactions. Aside from that don't waste clock cycles and potential problems. Even on the file server, as long as you do not use it as a desktop, and you require clients/workstations on the network to have anti-virus there's a low likelihood of there being a virus problem.
Your firewall your NAT router with stateful packet inspection should be able to handle external intrusions to your internal network. Run it on your clients if you want, and run it on any windows based servers or any servers exposed directly to the internet (say a web server), but it's not necessary to run an application host based firewall on your servers.