First note that there is the book of Vogan (Representation Theory if real reductive groups) which discusses the case of $SL_2(\mathbb{R})$ on a very basic level. I think this is a good start. In my opinion the remainder of the book is not very accessible.

However for the whole theory I would recommend that you first look at the notes from the conference "Computational Theory of Real Reductive Groups"

http://www.math.utah.edu/realgroups/schedule.html

Especially the notes from Adams and Vogan give a good introduction to the subject, don’t require much background and both are filled with examples.

In which direction to you want to go? Do you "really" want to do representation theory or are you planning to do geometric representation theory?

First note that there is the book of Vogan (Representation Theory if real reductive groups) which discusses the case of $SL_2(\mathbb{R})$ on a very basic level. I think this is a good start. In my opinion the remainder of the book is not very accessible.

However for the whole theory I would recommend that you first look at the notes from the conference "Computational Theory of Real Reductive Groups"

http://www.math.utah.edu/realgroups/schedule.html

Especially the notes from Adams and Vogan give a good introduction to the subject, don’t require much background and both are filled with examples.

Finally I should add Knapp's Book: Lie Groups Beyond an Introduction. At least the structure of real reductive groups (max. compact subgroups and so on) is discussed there.

In which direction to you want to go? Do you "really" want to do representation theory or are you planning to go to geometric representation theory?