Velocity is a vector quantity with both a magnitude and direction. The magnitude of velocity is speed. During the Earth's nearly circular orbit, the speed changes little, but the direction of motion changes, and this change of direction is the acceleration. The Earth is always accelerating towards the Sun.
When we speak of velocity one should mention the frame of reference: we measure the velocity of the Earth wrt the the Sun. The sun is therefore by definition "at rest" (we should actually use the centre of mass of the Earth-Sun system, an also include the gravitational effect of the other planets, I pass over these details) This frame approximates an inertial frame of reference
Mars doesn't stand still relative to either the sun or the Earth. We can use a frame of reference in which Mars is at rest, but this is then a non-inertial frame, it rotates, so in this frame both the Earth and Sun are moving, and accelerating due to the non-inertal motion of the frame.
If we (magically) remove the sun (but not the frame of reference in which the sun is at rest) then there will be no force on the Earth, and it will move in a straight line at constant velocity, relative to the frame in which the sun was at rest. It would move at the same speed as its orbit, about 100,000 km/h (in that frame of reference)