I just flew today and I was practicing touch and goes at Trenton Mercer and I am having major trouble staying on the centerline of the runway during landing, Takeoff roll, Takeoff, and departure leg are all places I have trouble, I think most of my problem is from keeping my eyes fixed on the instrument panel. Any suggestions or criticism would be much appreciated. Please post anything that you think will help!
The only instruments you ever need to look at at this point in your training are the airspeed indicator and the altimeter. It'll take a couple of minutes for you to get to pattern altitude so you don't need to look at the altimeter right away, and once you get stabilized in (and trimmed for) the climb, your airspeed should stay relatively constant on its own (if it doesn't, try letting go of the yoke and see what happens... If there's a significant pitch change, you were holding pressure on the controls and you weren't trimmed properly.)
That's the long way of saying look out the window.
After takeoff, it's not very important, IMHO, to be holding the runway centerline, especially once you're past the end of the runway. Keep your wings level and hold heading once you're climbing. If there's a crosswind, you should be holding aileron into the wind throughout the takeoff roll - Same pressure, less and less deflection as you accelerate. Your upwind wing will dip into the wind upon liftoff, and I generally find that having that wing dip followed by leveling off once I'm in the air ends up giving me roughly the crab angle I need to track the centerline. After that, I just hold heading (keep the wings level) and don't worry about the centerline - It's behind me.
As for landing, pretty much everyone has trouble landing left of centerline at some point, which I think is an artifact of trying to put the nosewheel on the centerline. Next time, try putting the centerline between your feet and having your body on the centerline, and you should find yourself right on the paint.
If you're landing both left and right of centerline, it may be a matter of your own tolerance, especially on wide runways. Don't let yourself land off the centerline, and you should do fine.
Hope this helps!