Here is one that has not been mentioned (I think). Is the nosewheel pushrod flexing too much? If the pushrod is flexing inside the fuselage, it can allow the wheel to turn when you don't want it to. Most trainers have the pushrod inside a plastic tube, so glue a piece of wood between the inside wall and the plastic tube about 2 or 3 inches from the servo. Be careful you don't get glue on the metal rod. Make sure it can still slide freely.
Something else occurs to me. I have seen planes that were wild on takeoff because the nose of the plane sets higher than the tail. As the plane accelerates down the runway, the tail will lift first. This has the effect of the plane rolling on just the nosewheel before it has quite enough lift to fly. With only one wheel on the ground, the plane can weave around quite a bit. For this, you can try bending the main gear down a bit (assuming they are the standard wire gear), or put larger main wheels on, or a smaller nose wheel.
If none of these works, it may just require more practice as has been said. Make your instructor takeoff the right way- so you can see if it's the plane, or just you. Across the runway, that's cheating!
