I'll try to answer your questions to the best of my ability, but I don't claim to be an expert when it comes to some VNC situations.
1) What would the internet setup be for the machines in this situation? Would you need a separate internet connection for each machine? Could they share the same internet connnection, using a router?
In my case, I have all the computers connected to the same router. In a home environment, just hook up your cable/dsl up to a router, and hook up into the router at both stations. I'm not sure what results you'd get with wireless. I would imagine that it would work, but it might not be the quickest, or most stable/secure method. I've also not tried hooking the two machines up to one another with a twisted pair ethernet cable. I would stick with a wired router, if possible.
2) Are there any security concerns of others gaining access to either machine in this kind of setup? What about doing online banking, brokerage, etc., via the server machine?
I would always exercise caution when doing these activities. Being on your own (hopefully secured) network should cut down on the chances of your personal info being snatched up, but as always use your best judgement (especially on a shared/public computer). Start with a solid password, and make sure to plug any network security holes. Of course, if you're doing banking on your "master" computer, and just accessing the "slave" computer through ARD, then I would imagine you're no more at risk than if you were using your computer without ARD active, since you're not enabling control of your "master" workstation.
I have not experienced any security issues, however it is certainly a valid concern. I'll try to find some more concrete information for you on the topic.
3) How difficult would it be for a beginner to VNC to set this up? I saw a demo of Apple Remote Desktop and it looked a little complicated. Any recommendations?
I've used Apple Remote Desktop to administer several computer labs, as well as to keep track of students' activities while in my classroom. ARD 3 is incredible! It's pretty straight forward to set up. There's a first rate tutorial at Apple's website:
http://www.apple.com/remotedesktop/tutorials/easysetup.html
As far as other VNC solutions go, I'm not very familiar with Chicken of the VNC, but I have used LogMeIn for a while now and, in addition to it being cross platform, not restricted to a local area network, and being free, it's a breeze to set up. Just follow these steps:
https://secure.logmein.com/support_settingup_lmi.asp
Please feel free to PM me, or let me know if you have any questions. I also welcome any other suggestions/criticisms/corrections from anyone out there that might have more experience than I.
Hope this helps!