I think I can clarify something for you. The Theory of Relativity is named that because everything is relative. What this means, is that if you were motionless in space and you saw a comet go by, it would be just as true to say that the comet was not moving and you went by it. Your speed must always be defined as relative to something else. Here on Earth we usually define speeds as relative to the Earth, and we define the Earth's speed as it orbits the sun in relative to the sun.
It's a hard concept to accept at first, but there is no "true" or "real" answer to which object is moving, it's all relative to where you are measuring from. For example if we had a small planet, and a spaceship out in interstellar space and we launched from the planet and moved forward at 99% of the speed of light people on the planet would observe our mass as increasing, and if they could see clocks on our ship they would see them moving slower. However, we would measure the exact same effect on the planet, as if we had remained still and they had accelerated to .99c (because both answer are equally true). If there were an ray of light passing by both us and the planet (and anyone else) would measure the speed as exactly the same. Even as we were moving at 99% the speed of light if we turned on our headlights the light would pull away from us at the exact same speed as if we were still (because again relative to ourselves we are).
Now in the above example you could still measure relative to the other stars and see that the planet was not moving relative to them, and you were, thus it would be tempting to say you were really moving they the planet really was still. If you took all the other stars away, and the planet how would you know you were moving, you wouldn't, because relative to your only reference point (yourself) you are not moving. There would be no way for you to detect the increase in mass from moving because all your instruments would have gained mass as well. After many generations on our spaceship, after all who knew the setup for the experiment were dead, if a comet passed by would we say we moved past it, or it flew by us? Both would be equally true, it would just depend on your point of reference.
Rest mass is the mass of an object while no accounting for any gains due to Relativity. You may think you could detect the gains when you began to move, but you would not, at least not if you and your instruments we part of the mass.
I hope I explained this well, I'm not an expert, but I had the same thoughts as you before (that you could tell if an object was moving by detecting gains in mass, and thus could say for sure that an object was truly motionless if it's mass = it's rest mass).
