A nice blog post published by ESRI developers team at their blog which contains a point to start materials for any developer wishes to start ArcGIS server  development using web API.

ArcGIS API for JavaScript

• Getting Started – Adding a Map (Copy the code and try it yourself)
• Building Mashups using the ArcGIS JavaScript APIs (Free, Self study, Virtual Campus)
• Building Web Maps Using the ArcGIS API for JavaScript (Instructor Led Course)
• Add a Map to a Web Page in Three Simple Steps (Article)
• Quick Intro to the ArcGIS JavaScript API (Intro Video)
• An Overview of the ArcGIS JavaScript APIs (DevSummit 2009 Video)

ArcGIS API for Flex

• Getting Started (Copy the code and try it yourself)
• Building Rich Internet Applications with ArcGIS API for Flex (Free, Self Study, Virtual Campus)
• Building Your First Rich Internet Applications with ArcGIS API for Flex (DevSummit 2009 Video) 
• ArcGIS API for Flex, Basics for Developers (Intro Video)
• Creating Your First Application using ArcGIS API for Flex (Two Minute Demo Video)

ArcGIS API for Microsoft Silverlight/WPF

• Getting Started Hello World App and Video (Copy the code and try it yourself)
• Introduction to the ArcGIS API for Silverlight (DevSummit 2009 Video)
• Interactive Samples (Try it live, copy/paste XAML, C# or VB.NET)
• Silverlight Toolkit (Open source for mapping widgets)

ArcGIS Server Services

• An Overview of Authoring and Publishing Services (Desktop Help)
• Creating and Publishing Web Maps (Desktop Help)
• ArcGIS Server REST API (Reference Doc) 

ArcGIS Developer Blog