Sharepoint Development

Sharepoint Developer Skillset:

The .NET framework
ASP.NET
C# and/or VB.NET
Web services
ADO.NET
XML/XSLT
Windows/IIS security
SQL Server
CAML

Online Microsoft eLearning Training
A great starting place is to take an online course.
WSS Development
MOSS Development

Read Books
Here's some great books on SharePoint development. There are plenty more available at the online book stores so make your own choice.
Inside Windows SharePoint Services 3.0
Inside Microsoft Office SharePoint Server 2007

Join the Discussion and Ask Questions on the MSDN Forums
This is a great place to search for answers, or to ask questions yourself, or to answer other people's questions. The SharePoint Developer and Programming forum is pretty active.
SharePoint Development and Programming Forum

Watch WebCasts
For WSS (the basic SharePoint API stuff) there are many on MSDN under Getting Started and under Learn.For MOSS there's also Getting Started and Learn material.

Spend time on MSDN
There are separate sections for WSS and MOSS so you need to go to both.
For WSS - http://msdn.microsoft.com/sharepoint -
For MOSS - http://msdn.microsoft.com/en-us/office/aa905503.aspx

References:
An introductory talk that Paul Andrew gave at SharePoint Connections, Spring 2008
http://blogs.msdn.com/pandrew/archive/2008/04/21/sharepoint-connections-talk-on-visual-studio-2005-extensions-for-sharepoint.aspx

Online Microsoft eLearning links are here
https://www.microsoftelearning.com/catalog/developer.aspx

More developer resources here http://www.microsoft.com/sharepoint/learning/resources.mspx

Microsoft Developer Evangelist Lynn Langit:
http://blogs.msdn.com/socaldevgal/pages/sharepoint-2007-developer-resources.aspx

SharePoint Development - http://www.microsoft.com/click/SharePointDeveloper/ (redirects from http://mssharepointdeveloper.com as well)

An introductory talk that Paul Andrew gave at SharePoint Connections, Spring 2008.
http://blogs.msdn.com/pandrew/archive/2008/04/21/sharepoint-connections-talk-on-visual-studio-2005-extensions-for-sharepoint.aspx

Online Microsoft eLearning links are here:
https://www.microsoftelearning.com/catalog/developer.aspx

More developer resources here:
http://www.microsoft.com/sharepoint/learning/resources.mspx

SPDisPose Check Tool -
http://blogs.msdn.com/sharepoint/archive/2008/11/12/announcing-spdisposecheck-tool-for-sharepoint-developers.aspx

SharePoint MCA Masters - http://blogs.msdn.com/sharepoint/archive/2008/11/10/introducing-the-microsoft-certified-master-and-certified-architect-for-sharepoint.aspx

SharePoint Patterns & Practices - http://www.microsoft.com/downloads/details.aspx?FamilyId=C3722DBA-6EE7-4E0E-82B5-FDAF3C5EC927&displaylang=en

VS 10 support for SharePoint “14” Development - http://blogs.msdn.com/sharepoint/archive/2008/11/10/visual-studio-2010-tools-for-sharepoint-announced-at-teched-emea-developers-2008.aspx

SharePoint SP2 in the Feb – April timeframe - http://blogs.msdn.com/sharepoint/archive/2008/10/24/prepare-for-the-upcoming-office-sharepoint-server-2007-and-windows-sharepoint-services-3-0-service-pack-2.aspx

Microsoft SharePoint Services & Azure - http://blogs.msdn.com/sharepoint/archive/2008/10/27/pdc-2008-announcing-azure-services-platform-and-microsoft-sharepoint-services.aspx

Cumulative Update Strategy - http://blogs.msdn.com/sharepoint/archive/2008/09/29/announcing-august-cumulative-update-for-office-sharepoint-server-2007-and-windows-sharepoint-services-3-0.aspx

Infrastructure Update - http://blogs.msdn.com/sharepoint/archive/2008/07/15/announcing-availability-of-infrastructure-updates.aspx

SharePoint Best Practices - http://blogs.msdn.com/sharepoint/archive/2008/08/29/announcing-the-sharepoint-best-practices-series.aspx

SQL 2008 Support - http://blogs.msdn.com/sharepoint/archive/2008/08/15/sql-server-2008-support-for-sharepoint-products-and-technologies.aspx

Virtualization Support - http://blogs.msdn.com/sharepoint/archive/2008/08/18/update-on-virtualization-support-for-sharepoint-products-and-technologies.aspx

Blogs:

Paul Andrew Blog -
http://blogs.msdn.com/pandrew/archive/2008/05/01/getting-started-with-sharepoint-development.aspx

Arpan Shah Blog - http://blogs.msdn.com/arpans/

Microsoft Developer Evangelist Lynn Langit:
http://blogs.msdn.com/socaldevgal/pages/sharepoint-2007-developer-resources.aspx

http://blogs.msdn.com/gregmcb/

http://blogs.msdn.com/mikefitz/archive/2005/03/15/396176.aspx