In regards to the right hand rule, given Earth's electric and magnetic field, in which "direction" would a particle go?
The field lines of earth's magnetic field go from south pole to north pole. So they are parallel to the surface at the equator, and perpendicular at the (magnetic) poles (going vertically into the ground).
Now, what happens to a moving charge in a magnetic field? A particle moving parallel to the field lines will not feel any force, however when it's moving perpendicularly to the field, it will be deflected according to the right hand rule. Note that the direction of $\vec v$ will change, but not the magnitude.
(The source of this behavior comes from the cross product in the formulation of Lorentz' force: $F_L = q \, (\vec v \times \vec B)$.)
Say your thumb points into the direction of the field line, your index finger in the direction of the particle movement. Then it will be deflected in the direction of your middle finger. Now turn your hand such that your index finger points into the new direction. You will see that the particle will describe a circle around the field line.
An incoming particle, say an electron from the ion wind of the sun, will in general have both velocity components perpendicular and parallel to the field. It will be moved away while describing the circle, and thus fly on a corkscrew-like path.
By the way, this is also the source of the northern lights (aurora borealis):
Because of the circular motion, the particles a) need much longer to reach the ground, and b) loose energy due to syncroton radiation. In this way, earth's magnetic field protects us from certain types of cosmic radiation!