It sounds like the three-way solenoid operates as it should. Its movement from back position to middle position should happen when you touch gas pedal slightly - this is governed by the throttle switch and can be adjusted, but I would not touch it for now. Then it goes forward when you press kick-down.
On Euro car with no emission control and no AC: if you have power with ignition turned on on the black/red (or black/red/green) wire (the power then is from fuse 5) and if you have ground on the red/green wire when the forward or backward gear is selected while engine is running (need two persons maybe to check it) - these may be wires to CSS to prevent engine from stalling when a gear is engaged (the ground is delivered by either of the two hydraulic switches on both sides of the tranny). But it may be different in your case if you have an US car.
To me it sounds like your modulator pressure may need to be slightly adjusted or the throttle switch may need to be adjusted - but see what the more experienced Members are saying.
However
Even though I am not a mechanic, I know that for the tranny to work properly, as it is governed by vacuum that engine is producing, the engine needs to be in tune. Ignition timing, dwell, valve clearances, idle adjustment - all that may affect shifting if out of tune. So I htink it needs to be made sure all this is correct before getting to tranny work fixing.