My favorite text is chapter 8 in Joseph Wolf's book "Spaces of constant curvature". I have a copy of the 5th edition, published by Publish of Perish, but a 6th edition by AMS Chelsea has recently come out. I don't claim it is an easy read, you need to work a lot on the details, but it gets to the point very efficiently. Perhaps it is fair to say that one can use it as a guide and complement the arguments as needed using the books of Helgason and Loos (2nd volume).
In particular, the classification of symmetric spaces is done in a rather elementary way, up to the case of involutions of $E_6$ which requires a bit of theory of roots (this part is best looked up in Loos' book).