You can easily make and program a robot arm without knowing anything about kinematics. However just like anything else in engineering, if you understand it your robot will work better and cost less.
You could if you want search for a preformulated equation matching the type of robot arm you want. You wouldnt have to understand how to derive the equation, just know how to copy and apply the equation.
Quick primer:
Forward Kinematics is if you know all the joint angles and positions of your robot arm, at what point in space would the end effector (gripper) be?
Inverse Kinematics is if you know the point in space you want the gripper to be, what angle would each joint need to be at? Inverse kinematics is obviously more useful, but much much more complicated to calculate.
In all situations, the fewer the joints, the easier to model mathematically. 4 DOF is therefore much easier.
As for books, I have read only one, so cant really recommend it. All of the books are heavy in math, so I wouldnt really recommend any
Matt Mason is a professor in robotic manipulation that I once studied under. This guy is top in the world on this stuff. He argues that a robot isnt intelligent if it cant juggle balls . . .
Anyway check out his homepage, has some useful stuff on it (such as class lecture notes):
http://www.cs.cmu.edu/~mason/
