How far can something travel in a straight line? Suppose you have an object some distance from you and moving at a velocity different to the Hubble velocity you'd expect at that point. How does the motion of this object change with time? Does it travel in a straight line (i.e. geodesic), and if you wait long enough how far will it get?
 A: As for the straight line, yes. All objects will continue moving along geodesics (a straight line in curved-space but sometimes a curved line in straight-space) if there are no external forces acting on them. Unless, by different velocity you mean the direction is not entirely radial to us. In that case, the expansion will cause the object's path to appear to "bend" away from us, but this is still a geodesic for the object.
As for how far it will get, this depends on your distance measurement. In proper distance (the distance you would measure with a ruler), the object can get infinitely far away if you wait an infinite amount of time. There is no limit in proper distance. There is also something called comoving distance, which factors out the expansion of the universe. Two points that are only getting farther apart because the universe is expanding remain the same distance apart in comoving coordinates. In this distance measure, there is a fixed distance that the object can get to. The image below shows the maximum comoving distance that we can reach at different velocities.

The blue, purple, green, and black lines are 0.5c, 0.8c, 0.99c, and c respectively.
Today, comoving distance equals proper distance. So the distances you see on that graph are essentially the maximum today-distances you could reach (even though the actual distance travelled would be substantially larger). Anything beyond those distances are unreachable from where we are now. Anything at those distances would take 70 billion years to reach at the given velocities.
So you see, an object would travel along a geodesic path and a ruler would measure it going an unlimited distance, but there is a limit on how far something can travel on a comoving scale. And that is the more useful thing to know.
