Chain skipping on new freewheel and chain setup - What went wrong?
sounds like your chainrings are worn.  Look at the rings and see if they are "shark fin" shaped rather than symmetrical - a google image search for "worn out chainring" will show examples including some really, really worn out rings.  The old worn out chain likely wore out the rings as well.

I've had exactly what you're describing on a couple bikes and it was remedied with a new chainring up front.