Say we have an object completely unaffected by the effects of gravity/velocity. Is there a way to measure the passing of time for this object? Since time moves slightly slower for us on or near bodies of mass & with movement/changes in velocity, is there a reference point or baseline for the movement of time?
In relativity there is no standard-clock that tells you which time is "right". That's the point about relativity. There is no need for a absolute reference to compare with. Everything is just the way you observe it (that is, relative to you). Things may slightly differ from observer to observer but the qualitative behaviour stays the same just as classical mechanics stays the same when you "look at it" from different directions (that is under a static rotation).
You may introduce something like inertial frames for which the effects of gravity are absent, however, this should be regarded as a histroical approach, because all we measure is in an approximately inertial frame. If we wouldn't be restricted to our planets surface we might have gotten the general covariant equations beforehand and have never introduced something like inertial frames.
There is no reference object that transcends all inertial frames of reference. Everything in this universe has an inertial frame of reference, and none of them are privileged. If there were any object that existed independently of the relativistic effects of acceleration/gravity or of observer movement, then theoretically it could provide a reference to which clocks could be set. However, no such object has been found in this universe.
The speed of light in a vacuum is independent of gravity and observer movement, but it doesn't provide pulsations to which clocks may be synchronized, and it is a scalar quantity rather than an object.
As the other answers point out, there is no "right" or "absolute" frame of reference for measuring time. But that does not answer your question:
Is there a way to measure the passing of time for this object?
It turns out there is. Oscillations in the orbits of electrons depend only on the material, and in fact our clocks measure the passing of time by counting the oscillations of particular kind of atoms. So, if you knew what material the object was, and were able to measure the oscillations in its atoms, you could in principle be able to measure the time passing in the neighbourhood of the object.
But definitely, the rate you would observe from the lab would depend by the relative speeds, and positions wrt gravitiational objects etc just as given by relativity.