A new chains on an old, worn cassettes or freewheel is the classic cause of chain skipping. Despite the fact you think the cogs aren't worn, they really are, particularly the smaller ones. Insatall a new freewheel and your problems should be over.
Chainrings last a lot longer and usually don't skip with a new chain unless they are very badly worn. Despite being softer than cogs chainrings are larger and have more teeth to interface with the chain and, most importantly, the chain goes onto the chainrings under tension so it tends to do what it's told.