You could use several different workflows for the type of work you're attempting. It's really up to you to decide which path to take. I'll offer my 2-cents from an ArcPad / GPS Analyst user's perspective.
The main thing that you're missing out on by using shapefiles and TerraSync is the ability to check out and check in your data from ArcMap. Additionally, creating data dictionaries seems like added work, especially if you're planning on teaching students the in's and out's of ArcGIS Desktop.
1. Storing all of your geographic data in a geodatabase (a personal GDB if you're planning on using GPS Analyst) is definitely the best approach. With feature classes you can take advantage of (among many other things):
(A) Attribute Domains. There is no need for data dictionaries if you set up coded value domains within the geodatabase. Coded value domains automatically get transferred to ArcPad so pick lists appear when creating new features. Personally, I feel that this is an incredibly important aspect of ArcGIS Desktop that all GIS students should learn. Plus, you can take advantage of these domains in ArcGIS Desktop as well.
(B) Check out / check in. With shapefiles, you can take them into ArcPad and edit them, but then there is no check in process available. You're left with an updated version of the shapefile on your mobile device and an outdated copy on your desktop. It's a constant copy / paste / overwrite method this way. With feature classes, you can check out to axf file (for disconnected editing) and then the edits you make to the files are checked right back into the geodatabase. New features are added, modified features are changed accordingly, and deleted features are removed.
(C) Relationship classes. If you have a relationship class setup in ArcCatalog these tables will go over to ArcPad. You can add new related records or modify existing records. Again, these modifications are automatically made within the geodatabase upon check in.
2. GPS Analyst / GPScorrect are pretty comparable to TerraSync and Pathfinder as far as the workflow for post-processing goes.
Really, once you have the GPScorrect settings all set up on the mobile device, it's just a few clicks within ArcGIS Desktop to post-process your data. Once you've checked the data back in (and waited for the day's base station data to be uploaded) it's one wizard that takes you through the post-processing. After that, all of your data is corrected and stored within the geodatabase.
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If you create your geodatabase and feature classes before you start your project initially, and only use one software suite to create and edit your data, it's going to make everything much easier down the road. I've seen issues from time to time with shapefiles created in TerraSync causing issues in ArcMap. Projections, field schema, etc., created or modified in TerraSync all tend to be slightly different than what ArcGIS Desktop creates. From my experience, it's best to create and modify data in only one software suite.
Hope this helps.