Accumulated charge in a grounded conductor 
My textbook states:

A charge $+Q$ placed near a grounded conducting sphere will cause negative charge to accumulate on the side closer to $+Q$.

However, it also states that the potential ($\varphi$) on and in a grounded conductor is zero. 
I don't understand how these statements can both be true at the same time. An accumulation of charge will create a potential. The non uniform charge distribution means the potential will be different at different points on the sphere.
 A: Before earthing
See, here the external field due to that positive point charge is non-uniform and non-zero at the surface of the sphere. So there is bound to be some non-zero (in this case, positive) potential on the surface of the sphere. And since the sphere is a conductor, it needs to have a constant potential at every point on its surface. However the field of the point charge is non-uniform and so is the potential generated due to this field. Thus to make the potential everywhere exactly equal, there is a non uniform induced charge distribution on the surface of the sphere.
After grounding
After grounding, we have allowed a pathway to charge the sphere. As soon as the sphere is earthed, electrons flow from the earth to the sphere's surface. Why? Because when you want a zero potential at the surface of the sphere, however the point charge's field creates a positive potential there. Thus to exactly cancel this positive potential, a negative charge is induced on the sphere's surface which would then create a negative potential at the surface, and so the net potential will cancel amd become zero, as expected.
The reason behind the non-uniform charge distribution is the same as before. Because of the non-uniform external field of the point charge, the sphere also needs to be non-uniformly charged such that the net potential at every point on the surface of the sphere is zero.
But what about the inside?
When the sphere acquires the above mentioned charge distribution, coincidentally the field inside the sphere as well cancels out (rigorous treatment requires an even larger answer and, of course, familiarity with the math used). Thus when you go from the surface to any point inside the sphere, you face no electric field and thus you have to do no work to move your test charge and thus the potential everywhere on and inside yhe sphere is constant.
