To move many keyframes at once you can use the Dope Sheet editor instead of the Graph Editor.
In the Dope Sheet editor all actions are represented by little diamond markers. In the above screen shot the column of markers on the right (colored) are selected, and the ones on the left are not selected. Selecting, deselecting, grabbing, etc in this editor works the same way as in Blender's other editors.
Notice the order of hierarchy: The top row is for all objects in the scene. The second row is for just the "Cube" object. The third row "CubeAction" is for actions involving the object, such as LocRotScale transformations (but not for things like Material animation). The fourth row "LocRotScale" contains the actual animation keyframes and if you expand it you will see the individual rows/tracks for Location, Rotation, and Scale for each of the X/Y/Z axes.
So if you want to move just one axis you would expand "LocRotScale", find "X Location" for example and move just its individual marker. But to do what you described - moving all axes at once - you can just leave "LocRotScale" collapsed and drag that marker. All of the animation data will be moved together.
If you want to adjust just the ease of the curves specifically and not their position, you can select the keyframes in the Graph Editor and scale them along only the X axis. Their bezier handles will be moved closer to or farther from the keyframe points, thus altering the ease.
As a variation of this you can try scaling with the Pivot Point set to Individual Centers. You may find that scaling and rotating from individual centers gives you more control than simply scaling along X.