I have a now stay at home mom and two teenaged kids (or almost anyway). Everyone wants to use the one computer in the house.
I thought about getting a couple of PCs and then networking them somehow onto the internet line.
But I also thought about a mainframe with terminals around the house. I believe I could get one pretty reasonably (a friend was in the business and has a stash of them) but wonder what the downside might be.
I think you mean "server", rather than mainframe. Mainframes generally cost a couple million bucks these days and require a large amount of infrastructure. Not to meantion, few if any run Windows, MacOS or Linux. z/OS, AIX, etc are fairly common. Servers are similiar to PC's in most regards, except they're meant for multiple users, being left on all the time, longer lasting components and such. Rack mountable ones look like metal pizza boxes.
For a home server, you'd be fine with a regular PC with a few extra hard drives.
You have a couple options.
Centralized - a heavy server and lots of 'thin clients'. Thin clients are a nice way of saying "PC's that suck". It's a regular PC with no extras and maybe not even a hard drive. That'd be fine if you just wanted to browse the web and maybe some light word processing. It can be a pain to set up but usually doesn't have issues afterwords. Thing is, if your server goes down, your thin clients will too.
Decentralized - a light server and normal PC's for around the house. Get unexpensive computers and do most of the work on them. Just store stuff on the server. If your server or any one computer crashes, the rest still work just fine.
A router can manage your networking. A decent LinkSys home router can do DHCP, NAT and act as a simple firewall. If you get one with a couple ports or wireless, you can network your computers through it as well to make them talk to each other.
The more tech savvy thing to do is to buy (or "obtain") a copy of server 2003 and a handful of remote desktop licenses... or use the bunches of other software packages out there for terminal access. The beauty of this is that the remote terminal isn't really doing any work... Upside: cheap hardware (PII-500 would work just as well as a brand new box for a terminal), ultimate central control. Downside: harder to configure, multimedia and gaming wouldn't work.
There's really no need to pirate Server 2003 R2. Microsoft makes a version called "SBS", which is a lightweight version of the regular server software. Stands for Small Business Server. Server 2003 comes with 5 built in CAL's (number of users or devices allowed), before you need to purchase additional terminal services licenses. It's not that expensive.
Going with a couple light PC's for normal day to day stuff shouldn't be too expensive either. If you wanted multimedia or gaming, you could have one higher end machine for such things.