In After Effects I could grab a handle on a curve, hold down a modifier key, and only one side of the curve would be affected.. Is this possible in Blender in the Graph Editor?
That depends on how the handles are set. By default, on a Bézier curve the handle type is set to Aligned, that's why both sides of the curve are affected. If you select a keyframe and press V you get the Set Keyframe Handle Type context menu. There you can choose Free.
If you want these Free handles temporarily to affect both sides without changing it back, you can select them both and use R to rotate or S to scale them together.