Seeking ArcGIS Desktop tutorials?
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Ensure you have ArcTutor installed (comes with ArcGIS Desktop install)
Start doing the tutorials.
Id also advise paying a trip to the ESRI Virtual Campus. - Plenty of free training on there to get you started.
Go to video.esri.com for inspiration.
Understanding your data (projection, attributes, formats) then you can using ArcGIS in some quite amazing ways..
General Usage : For a good basis of understanding a geodatabase. http://webhelp.esri.com/arcgisdesktop/9.3/pdf/Building_Geodatabases_Tutorial.pdf (yes 9.3 but still worth the look - good starting point)
Focus on: Subtypes and Domains (and validation spatial data) Relationships between objects. Topology - very powerful in controlling spatial features and their relationships.
for ArcGIS 10 http://help.arcgis.com/en/arcgisdesktop/10.0/help/index.html#/An_overview_of_geodatabase_design/003n0000000z000000/
If have access to MS Visio you can see the structure of the Geodatabase http://resources.arcgis.com/content/data-models
For really fancy stuff (making gis data look nice) look into Representations http://help.arcgis.com/en/arcgisdesktop/10.0/help/index.html#/What_are_representations/008t00000002000000/ which is tied to spatial data (keeping the topological) structure but represent the data in a more human (cartographic) form.
The most advanced and powerful part of ArcGIS is the Geoprocessing tools that can be automated to do serious amounts of analysis. http://help.arcgis.com/en/arcgisdesktop/10.0/help/index.html#/A_quick_tour_of_advanced_techniques_in_ModelBuilder/002w0000005q000000/
First of all, I would suggests, to start from help of Arcgis. You will find a lot of examples and tutorials.
For videos you can find some in youtube channel of esri and more in other channels:
http://www.youtube.com/esritv
You mension about Geodatabaseis, a good idea to start is the book of ESRI (i know is theoretical but sometimes you need it):
Designing geodatabases: case studies in GIS data modeling
and free from google...:
Designing geodatabases: case studies in GIS data modeling
Very good books with tutorials is:
GIS Tutorial 1: Basic Workbook
GIS Tutorial 2: Spatial Analysis Workbook
GIS Tutorial 3: Advanced Workbook
In some forums you can take some ideas:
http://www.cartotalk.com/