I have a shapefile or esri geodatabase file of Appraisal data that I need to join to two different appraisal data tables based on a single field and add the table fields to the original file. What transformer(s) should I use?

You have a few options for that. I use a FeatureMerger often when I'm joining two datasets based on a common attribute (ID). But you could also use a FeatureJoiner or a DatabaseJoiner and there's probably others. If you use the FeatureMerger, I would merge the first table with your shapefile and then run that output into a second FeatureMerger merging with your second table.

@olesonr​  The article Merging or Joining Spreadsheet or Database Data has a nice flow chart that can help you chose the most appropriate join tool from the many that FME offers.