Packt has some ArcServer books for $5 this week, Esri and http://geospatialtraining.com/ are two companies that could help with training. But getting access to ArcServer software is expensive, Esri EDN has the software for $1,500 but is yearly. A cheap option is to get an Esri Developer account and learn how to use ArcGIS.com to create your own services but it's not ArcServer. ArcServer is mainly a way to server data, and geoprocessing service.
The second option is programming to consume those services, for this the best option is JavaScript, using the ArcGIS JS API or Leaflet JS API, and now the Python Web API. These options are free to use and the web programming skills will enhance your resume. Leaflet is probably the easiest one to get started with and the Esri plugin allows you to use ArcServer services. Also you can take shapefiles and convert them to GeoJSON and use them as a data source. There is a ton of public ArcGIS REST services out there to get you started, Mash this up with some of your own GeoJSON or ArcGIS.com services and you can create some great web maps.
Using ArcServer, GeoJSON, or ArcGIS.com to create data sources is the easy part, Being able to create web map applications and use the data is a far more marketable skill..
