In the last couple of weeks I've been reading J.E. Humphreys' "Introdcution to Lie Algebras and Representation Theory" and after finishing Chapter 3 I had a chat with my TA about the topics*. He told me to read up on the examples of $\mathfrak{su}(2)$ and $\mathfrak{su}(3)$ to see an application of the theory to some simple cases. The suggested reading material are relevant sections from Georigi's "Lie Algebras in Particle Physics". I tried reading some chapters from Georgi's book today and I must admit that I don't understand anything that he writes. I can of course recognize similar patterns and formulas, but I don't see myself really gaining a better understanding of the concepts of roots, weights, etc. by reading his book.
I'm therefore asking for a reference that discusses in a bit more detail the concept of Weights, Weight spaces and the root system (root diagram and simple roots) of the two Lie algebras $\mathfrak{su}(2)$ and $\mathfrak{su}(3)$.
*) The topics covered up to this point where the necessary theory for the classification theorem of semisimple Lie algebras via Dynkin diagrams (aka Ideals, Nilpotency, Solvability, Root systems, etc.) and also Weights and Weight spaces.