By Newton's second law: F = ma (magnitude only)
When did I say mass == weight?
F(due to acceleration of the elevator) = m(person)*a(acceleration from elevator)
F(due to gravity) = m(person)*g(gravitational acceleration)
F(total) = F(due to acceleration of the elevator) + F(due to gravity)
=> F(total) > F(due to gravity)
Result: Person appears to weigh more.
Gravity is still acting on the person though, so not all forces are cancelled.
Gravity is not a force as you imply. There is a force due to gravity, which is usually called weight. The person can accelerate due to gravity, but gravity does not "act" on the person. It is the force due to gravity that is acting on the person.
Assume the acceleration due to the plane is a and that a equals g, the gravitational acceleration (Never really equal but for the sake of simplicity, let's say they are).
Assume no friction due to air.
Assume the mass of the person does not change.
Assume the plane is ascending.
F(due to gravity) == F(due to acceleration of the plane)
but the directions are opposite, so the forces oppose each other.
F(total) = - F(due to gravity) + F(due to acceleration of the plane)
F(total) = - m*a + m*g
F(total) = - m*g + m*g
F(total) = 0 N (Newtons)
Forces have cancelled each other out, and the person experiences weightlessness.
In reality, the forces do not evenly cancel each other out, but the resultant force is low enough in magnitude that the person feels weightless.