How do I avoid fibrous, dry beans?
"Fibrous" and "dry" are good descriptions for bean which haven't been prepared properly. Mostly, they have either not been soaked long enough, or haven't been cooked for long enough.

Normal times for beans are about 12 hours soaking in water and another 1-3 hours cooking, depending on the desired texture.
Also, sometimes beans can be cooked with about quarter/half a teaspoon of bicarbonate of soda, which softens them nicely.

That should be a start towards solving your problem. To answer your question, I think that white beans and pinto beans are a good start (you can make a chili, for instance), and that chickpeas are also a very good starting point. Cooked chickpeas are great as-is, and can be seasoned with cumin, salt and lemon juice for a nice snack.