When to use the single character vs double character form of a verb?
I'll try answering this (but be aware I'm just a learner).

The given examples are transitive vs. intransitive verbs:


说 vs. 说话.  Here, 说 is a transitive verb (i.e., it takes an object) whereas 说话 is an intransitive verb (i.e., it doesn't take an object):


  他说“你好”
  He says hello
  
  你为什么那样说话？
  Why do you speak that way?


We can alternatively think of 话 (noun) as what is being said (说).
吃 vs. 吃饭.  Again, 吃 is a transitive verb whereas 吃饭 is an intransitive verb, so 我吃饭面包 is incorrect, but 我吃面包 is correct.

Again, we can alternatively think of 饭 as the object being eating (吃).


(I also note that both 说 and 吃 can be used as intransitive verbs, e.g., 你别说了 and 他什么都没吃.)

学 vs. 学习

Another example is 学 and 学习 which are both transitive verbs.  So both of these are grammatical:


  我学汉语。
  我学习汉语。


In this case, which do we choose?  See this question: What&#39;s the difference between 学 and 学习?  It's somewhat a matter of preference, circumstance, etc.