To answer the title of the question, gravitational mass and inertial mass are identical and not just equal. This is the principle of equivalence.
When an object is travelling near speed of light, it's mass increase.
An object's mass (referred to as rest mass) is an invariant. i.e., it does not depend on an observer's frame of reference. Relativistic mass$^1$ i.e., $m=\gamma m_0=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$ is a deprecated concept. The total energy of an object with rest mass $m_0$ is $$E=\gamma m_0c^2\ \ \ \ \text{with rest energy}\ \ \ \ E_0=m_0c^2$$
As objects travel fast, will they become heavier too?
Will trains traveling fast weight more?
The concept of weight and mass are distinct. Mass is measure of inertia (resistance to force) and weight is the force exerted on a mass due to gravitational force. That is, $W=mg$.
But the question is interesting. First note that it is not as simple as saying it has a "relativistic gravitational weight, like $W=\gamma m_0\cdot g$", which is perhaps what you imagined and why you asked the question.
Gravitational fields arise from the stress-energy-momentum tensor $T_{\mu\nu}$ (see the Einstein field equations). So, since the object has an associated stress-energy-momentum due to its relativistic motion, $T_{\mu\nu}$ changes so as to change its gravitational field (increasing its pull on earth) and therefore the earth increases its force on it (it is slightly more complicated than that, but this should be roughly true). So it becomes "heavier" just not the way you thought.
$^1$ Relativistic energy is given by $$E^2=p^2c^2+m_0^2c^4$$ for an object moving with velocity $v$ and has momentum $p=\gamma m_0v$.
