Why observe these fascinating objects ? Well, there are plenty of reasons. The Budget Astronomer lists them as follows:

1. They are good targets for an urban or suburban location. Many of them are quite bright, and there are binaries visible all year long, whereas planets and the moon are not always up when you want to observe.
2. They are aesthetically pleasing. A nice binary pair with distinct colour differences can be very pretty.
3. They provide a challenge - some are faint, or very close, or have a high magnitude difference, and they provide a certain satisfaction.
4. Binaries give you an excuse to push the magnification on your scope to "crazy high" powers - above 600x.
5. You will never run out. There are thousands of binaries visible from any given latitude.
6. Binaries provide amateurs an opportunity to contribute to "real science" - there are more binaries than professional astronomers. Backyard observers can contribute by measuring the separation and position angle, which can then contribute to calculating orbits, which in turn helps us learn more about the mass of the stars, and helps to refine the model of the nature of how stars form and evolve.

The AAQ monthly newsletter includes many interesting doubles to observe month by month, in the Visual Observing section. There are websites which include several lists of  doubles worth a look. There are books which deal with double stars or include aspects of doubles and their observation.