Learn Dutch: Er loopt een paadje naar de wijngaard. - A small path leads to the vineyard.

This is exactly the vowel shift you are learning in this lesson! For a small group of nouns, the short vowel in the singular noun lengthens when it becomes a diminutive. So the short 'a' in pad becomes a long 'aa' in paadje. Other examples include glas becoming glaasje and gat becoming gaatje.

The English translation says 'leads', but doesn't loopt mean 'walks' or 'runs'?

Yes, lopen usually means to walk or run. However, Dutch uses lopen to describe the route, course, or direction of inanimate things like roads, paths, and rivers. When a path or road goes somewhere, Dutch speakers say it 'walks' there.

Could I just say Een paadje loopt naar de wijngaard without using Er?

You could, and it is grammatically correct, but it sounds less natural. Dutch strongly prefers using er as a dummy subject to introduce an indefinite noun (like een paadje, a small path) into a scene. It translates literally to: 'There runs a small path to the vineyard.'

What is the gaard part of wijngaard?

It is an old word for an enclosed yard or garden. You will also see it in words like boomgaard (orchard, literally 'tree-yard'). Combined with wijn (wine), it creates the word for vineyard.