To answer your questions:
Q: What software do I need to download and install? Server? Client?
A: There are a number of Subversion clients for Windows, including AnkhSVN that integrates with VisualStudio and one that is very popular with our clients, TortoiseSVN, which integrates with Windows Explorer.
On the server side, I'd recommend going with a hosted solution if your corporate policies allow. Just put "subversion hosting" into Google and compare the available providers. If not, I recommend installing the Windows binaries from the Subversion site.
Q: What do I need to install the software on? Windows 2003?
A: Subversion is available for all major platforms from the above link, so if you're a Windows shop, sure, Windows 2003 is fine. If you are heterogeneous, you may have a bit of an easier time configuring it behind Apache on Linux. Subversion is well-documented and has instructions for configuration on all platforms though.
Q: Should I have a dedicated server or is an existing file server adequate?
A: Again, I recommend hosting externally over hosting it yourself. Any hosting provider worth using will provide the following advantages:
*Accessible securely from anywhere via HTTPS (no need to configure corporate firewalls to work from home or a client site)
*Offsite backups (no need to set up new backup procedures for that dedicated server or file server in your office)
*Uptime (no need to worry about the box melting down or having some other issue that your team will have to deal with directly)
*Configuration (UI-based setup rather than having to learn server side configuration procedures)
If your company requires hosting it yourself, and there is sufficient space carved out on the file server, I would say it's not a problem to use an existing server. If you have a dedicated server handy, that will be better just from the perspective of not having to "bump heads" with other parts of the organization assuming it's a shared resource.
Q: How quickly will developers get up to speed? Is the integration similar enough to just carry on as normal?
A: There is a slight learning curve for Subversion, but Chapter 2 of the Red Bean book as well as the documentation for the clients explains how a regular work cycle goes. If your developers are familiar with CVS they'll be up to speed in no time. If they aren't, it may take a little while to get used to the notion that you don't have to lock files to work on them (Subversion has a locking mechanism but it's not designed to be part of the standard work cycle).
Definitely feel free to contact me directly if you have any other questions.