If the earth is moving with constant acceleration.
Then einstiens theory of reletivaty is surely meaningless. It states that the faster a body with mass goes the heavier it gets.
It does state that, yes- when an object attains a high velocity with respect to a given frame of reference, its mass increases from that frame of reference. Relativity is not meaningless, and is perfectly compatible with FE.
And once a body with mass travels at the speed of light it has infinate mass, which is impossible to have.
It may or may not be impossible to have that, but the Earth is not traveling at the speed of light. It is traveling very near the speed of light relative to some objects, and not very quickly at all relative to other objects.
First of all have we not accelerated enough to reach the speed of light? and if not then how fast are we traveling at the moment?
No, we are not traveling at lightspeed, and never will. Of course, it all depends on what you are measuring Earth's velocity relative to; relative to some reference points, we are approaching lightspeed, and from those reference points our acceleration would appear to be decreasing. Relative to other frames, we aren't moving particularly quickly, and still retain a constant velocity of 9.8m/s/s.
and secondly, if the earth and everything on it is accelerating then we would all be gaining mass as seconds go passed.
that doesnt happen.
Relative to some frames of reference, we are gaining mass- but only for those frames with respect to which we are approaching lightspeed. Relative to us, the Earth's velocity is zero, so from our reference point its mass and acceleration remain constant.