How would I get started with developing a custom web service in SharePoint?
Are there are tools I could use, or tutorials that would help me?
Below are a few links you may find useful. Unfortunately this is a very broad question, so it isn't possible to include all the details here.
MSDN Walkthrough for a basic web service:
A SharePoint 2007 web service walkthrough (with downloadable code):
A SharePoint 2010 WCF service walkthrough (with downloadable code):
The most important tool is obviously going to be Visual Studio. I haven't heard of any tools to specifically help you with this project, although there's a generally useful list of SharePoint Development tools here: