SQL Server 2016 full backup and log shipping and newbie questions As most likely in a lot of places, I find myself in the roll of "DBA" by default rather than by skill.

I currently am setting up new SQL Servers using SQL Server 2016 Standard Edition. I have been able to successfully set up the log shipping agents and schedules to where it took a full back up at start, and now does log shipping every minute, and does a restore on the secondary every 5 minutes. I have verified its working as expected by altering the primary database and, after the 5 minute boundary has passed, I can see the changes in the secondary database.

First question I have is, once the restore on the secondary has been done, can the transaction log files be deleted? 

Next question is, if I wanted to combine the transaction log shipping with a full/differential backup plan, is there anything out there that explains better about how to set this up. I have read multiple articles, but each seems to be missing one component or the other and its making it difficult for me to understand how these differential backup components work together.
once the restore on the secondary has been done, can the trn files be
  deleted?


Yes you can but you want to consider keeping files for certain period of time depending on your recovery point objective.


  if I wanted to combine the transaction log shipping with a
  full/differential backup plan


Full and differential backup will not break your log chain.  

Make sure you do not take any log backup outside log shipping. All log backups must be done through the log shipping. Because that will break your log chain and hence log shipping will fail.