Does a gravitational field get transformed when switching from one inertial frame to another? We know that electric and magnetic fields get transformed when switching from one inertial frame to another as per the rules governed by special relativity. What about gravitational fields?
 A: In general relativity the term “gravitational field” is a little ambiguous. There are at least three quantities that could qualify. The one that is most central to GR is called the metric. The one that best describes the curvature aspect of GR is called the Riemann curvature tensor. The one that is closest to the Newtonian gravitational field is the Christoffel symbols. 
In a very important sense, the metric and the curvature tensors do not change when switching from one reference frame to another. Just the components in the coordinate basis change, but the underlying geometric object is the same. 
However, in that sense, the electromagnetic field also does not change. Since you described the electromagnetic field as changing, you seem to be interested specifically in the change in components for different coordinate bases. The metric changes in the same way as the electromagnetic field, with the only difference being that the electromagnetic field tensor is antisymmetric and the metric tensor is symmetric. The curvature tensor transforms similarly, but it is a higher rank tensor so it is not as closely similar as the metric tensor. 
In contrast, the Christoffel symbols, the closest analog to the Newtonian gravitational field, change in a much different fashion. They are not covariant, so the change of components is more than what can be accounted for by a change in basis. For example, if a true tensor is zero in one frame it is zero in all frames (a zero vector has all zero components in any basis), in contrast, the Christoffel symbols may be zero in one basis but nonzero in another basis. In particular, inertial frames are those frames where the Christoffel symbols are zero. So, by restricting your question to inertial frames you are forcing the Christoffel symbols to be zero and hence to not change. 
Sorry the answer is not a straightforward yes/no. There is just a bit of ambiguity about what is meant by the term “gravitational field”. I would say that the “gravitational field” is the Christoffel symbols, so there is no gravitational field in an inertial frame. So the gravitational field does not change between inertial frames since it is identically zero in all of them. 
A: Yes, it does. For instance, if an observer measures a G-field as $g^\prime$ in a reference frame at rest with respect to the G-mass (planet), yet far enough from it, the observer who moves at $v$ perpendicular to the field lines measures it as $g=g^\prime/\gamma^2$, where $\gamma=1/\sqrt{1-v^2/c^2}$. If the motion is not perpendicular, i.e., for any arbitrary given direction, the calculations are slightly complicated, however, one can use three-acceleration by replacing $a$ with $g$. 
Remember that it is needed for both of the observers to be far enough from the planet, i.e., to be in the weak-field limit. Otherwise, GR amendments are needed.
