To expand on tomsrobots answer a bit.
If your desired arm end effector pose is far from the current, you should use an inverse kinematics solver to give you the corresponding joint angles. Typically, for high DOF arms, there are multiple solutions, so you should pick the best solution based on some metric. Minimum distance in joint space (i.e. configuration space) to the current or some known good configuration works well.
Depending on the complexity of your arm, the IK could be as simple as some trigonometry and conditionals, or require a full-blown IK library. See for example ROS MoveIt! and OpenRAVE. Standard robotics textbooks should cover the basics of what is involved here.
Then, you can move the arm there in a number of ways. One joint at a time, or all at once like you mentioned. These move the arm in joint space and the trajectory of the end-effector will probably not be what you want. But typically, for advanced motions you will plan a path in cartesian space and the arm will then follow the trajectory. But for for relatively open workspaces, and small motions, a straight line in cartesian space is typically fine.
Then the arm must follow the trajectory. If you used a motion planner, it probably gave you a trajectory in joint space.
Another way is to input your desired end-effector cartesian velocity into the arm's Jacobian, which will then give you the required joint velocities. The Jacobian is dependent on the arm's configuration, so you need to iterate. (i.e. get joint velocities, move a small amount, repeat.) Again, unless your arm is very simple, you probably want a kinematics library mentioned above to calculate the Jacobian for you. Note that you can often skip the IK and path planning steps and go straight to following a trajectory with the Jacobian if you are moving short distances, you know you are far from singularities, you have no obstacles in the workspace to get around, etc.