I'm not aware of any manuals with possible exercises. There are some textbooks floating around. Others may have titles. There have been titles mentioned in topics in the Assembler section of this forum. A Google search for PC/370 will turn up some textbook titles if you have the patience to go through some of the links, though these may be oriented towards the now obsolete PC/370 product. Similarly, a Google search for z390 may turn up some suggestions.
I started out doing things that looked interesting, but that was 40+ years ago, though the same exercises can still be done. Writing a program that lists a PDS directory in the same general format as the IEHLIST LISTPDS function is just as challenging for a beginner as it was for me in early 1969. Replicating the IEBPTPCH PRINT TYPORG=PO is another good exercise for a beginner; you'll learn quite a lot about READ/CHECK and deblocking data blocks, as well as using FIND ttrn,C to point to the first block of a member using the TTR you obtain from reading a directory.
One thing that beginners have trouble with is conversions from binary to text and from text to binary. Learn those conversions and learn them well; you will find you need them all the time. Don't make the mistake of using band aides like the Assist XDECO / XDECI macros. Use this opportunity to learn the ED instruction for simple conversions, this will be helpful later on when you may be faced with more complex conversions.
Doing these exercises will inevitably lead to errors and dumps. These will not be complex dumps, so use this opportunity to learn the basics of how to read a "simple" dump; this will be an extremely useful skill later on.