measure theory for dummies
Every textbook on measure theory that I've looked at has plenty of simple examples of the kind you mention (not with trees and apples, but simple nonetheless). What do you find lacking in the texts you've read?
Regarding your example $A:= \{\text{tree,apple,1} \}$, it's a mistake to ask what is the sigma algebra of this set (I hope I understand your question correctly here). There exist multiple sigma algebras of members of $A$: $\{ \emptyset, A \}$ is one, the power set of $A$ is another. Again, any of the standard references should make this very clear.
When I started learning about Lebesgue measure and integration, I found Taylor's General Theory of Functions and Integration very helpful (and still do). It moves slowly and gives lots of examples. It also has a Dover edition and so is very affordable.
If you're interested in an introductory text on measure-theoretic probability, I can recommend Rosenthal's A First Look at Rigorous Probability Theory. I would not consider this a textbook in measure theory proper, but it explains and makes use of the basic measure-theoretic concepts needed for probability and the exercises are not too difficult.
Addendum. I will stick with my original recommendations in light of your edit. You should keep in mind that, in learning mathematics, part of your job as a reader is to think of intuitive explanations and simple examples of the new concepts that are introduced. That's how one learns. Reading math is an active affair; you have to struggle with examples and exercises until the concepts become familiar. No matter how clear and simple your author may be, you'll never learn math by just passively absorbing a textbook.
Personally, I do not know of a book that simple. With that being said, Terrence Tao's An Introduction to Measure Theory is quite approachable and readable as an introduction to Measure Theory, assuming you have the prerequisite background.
More particularly, if you want simple examples, focus first on the Lebesgue Theory. It is more geometric and a bit less abstract, but it provides a firm base for the pursuit of abstract Measure Theory later.