I'm designing a plane, and I have a question. What is the best place for the center of gravity, relative to the center of lift? I've heard that it's best put just behind the center of lift, but I would like a second opinion.
The center of gravity (cg) should be right above or below the center of lift. Otherwise the airplane will start to pitch and never stabilize.
But you do not select the position of the cg, instead, you adjust the center of lift such that lift will act along the same axis as weight. This is called trimming the aircraft and is done by adjusting the elevator deflection (or the incidence of the stabilizer). Of course, you should select the cg position such that the available elevator deflection range will not only shift the center of lift to the cg, but leave enough margin for manoeuvring.
The next condition is to ensure proper natural stability. Once the tail surface produces more lift per area than the forward surface, the aircraft will be instable, so that any angle of attack deviation from the desired state will quickly increase. Therefore, the cg should be placed slightly ahead of the neutral point of the airplane. The neutral point is the cg location at which angle of attack changes will not cause a change in pitching moment. The neutral point of a single wing is near its quarter chord point (it moves forward from the quarter chord point with decreasing aspect ratio), and the neutral point of the whole aircraft is the mean of the neutral points of all components, adjusted for their lift curve slope.
A good first position for the cg is at the quarter chord point of the wing. This ensures that the tail surface will be only lightly loaded, but it might be that stability at this position is not satisfactory. To increase stability (and reduce pitch agility), move the cg further forward, and vice versa. If you want to optimize the aircraft's lift-to-drag ratio (and minimize trim drag), select the most backward cg position with still acceptable stability.
GA aircraft are normally very stable, and their cg position might even be close to the leading edge of the wing. Details depend on tail size, aspect ratios of all surfaces and the desired agility.