First off, a major point of confusion is regarding the name "SharePoint" -- it actually encapsulates many different products and technologies. I can only speak specifically regarding WSS/MOSS which is really the only thing you should concern yourself with if you want to begin experimenting with it.
WSS is Windows SharePoint Services (currently on version 3.0). It can be installed on any Server 2003 or Server 2008 operating system. It has no license fees aside from standard Windows Server licensing and is a great place to start getting into SharePoint.
MOSS is Microsoft Office SharePoint Server and it builds on top of WSS 3.0. It is not free. It has its own set of licenses depending on what you plan on doing with it (public internet costs more money for example).
To help answer the rest of your question, which is honestly outside the scope of this little post but I will do my best), I reccomend the following:
1) Get used to virtualization. If you can get your hands on a Server 2008 workstation and utilize Hyper-V that would be great. Otherwise a standard Vista box with MS Virtual Server installed would work as well. Any virtualization you choose is fine, but it's definitely recommended that you not install/develop SharePoint on your base environment.
2) Once you create a new virtual environment you are free to install Server 2003 or Server 2008 (WSS does not install on a client OS like Vista/XP/Windows 7).
Once you have your virtual Windows Server loaded up and at the desktop:
install WSS using this link:
a) If using Windows Server 2003, if you Windows Update, you should see a Server Role for Windows SharePoint Services. If you enable this role, it should install everything for you.
b) If using Windows Server 2008, there is currently no WSS Role, so you can install WSS 3.0 from the following link:
http://www.microsoft.com/downloads/details.aspx?FamilyId=D51730B5-48FC-4CA2-B454-8DC2CAF93951&displaylang=en
3) Install Visual Studio 2008. You'll also want a few tools to help out.
a) Visual Studio Extensions for SharePoint: http://www.microsoft.com/downloads/details.aspx?FamilyID=B2C0B628-5CAB-48C1-8CAE-C34C1CCBDC0A&displaylang=en
This addin will add some templates to help get started.
4) Lastly, there is a final tool called SharePoint Designer that may help you with debugging and just generally playing around. I believe it is a free download from Microsoft now.
Beyond this, I can only give you some resources to point you in the right direction. SharePoint is a massive platform. You can really take it a great number of directions depending on what you want to do. I have been using SharePoint for 2 years now and I haven't touched on many parts of functionality because the need never arose.
Before even playing in Visual Studio I would get used to work with the web interface. You can really play with a lot of things once you have a standard WSS site deployed (web parts, creating Lists, Document libraries, etc), custom pages, galleries, etc.
Resources:
Here are some of the blogs I follow from prominent SharePoint developers
1) http://www.andrewconnell.com/blog/Default.aspx (He has some great getting started tutorials as well)
2) http://www.harbar.net/
Lastly, I can add the fact that SharePoint 2010 looks to be a huge upgrade from the 2007 release (WSS 3 / MOSS 2007). It might be worth looking into some of those tutorials and developer blogs from MS. SharePoint is going to be much more integrated into Visual Studio 2010 as well which should help get developers started more quickly as well. Not to sound negative or deter you, but it may be less frustrating for you to wait until a CTP of SharePoint 2010 is released before you dive in. Getting started in WSS 3.0 is nothing short of very frustrating and very trying on your patience due to all the manual steps required before you can actually deploy something (let along upgrade an existing deployment to a new version).