I’m sure this is covered all over the place by just about everyone, but I can’t seem to find it.
If I have a box animated along its x axis, then duplicate it and slide it to the side (y) a bit, so they both animate along side each other, the duplicate always jumps back to the original position as soon as I move the time slider. Is there a quick easy way to have it stay off to the side (the new y position)? I tried Object > Make Single User, but that doesn’t work.
There are a number of ways. One is to change the Transform keyframes into Delta Transform keyframes. This keyframes the difference from the startpoint rather than absolute position values
See demo: http://screencast.com/t/5hWMRzuPIyT
Thanks! Exactly what I was looking for. 
And thanks for taking the time to screencast it so as to be very clear. Much appreciated!
You can also turn off the y-axis movements in the little check boxes (from expanding animations in the animation pane) so that the only animation is your x and z-axis movements (thus it won’t change the y position).