"Shade" vs. "shadow"
A shadow is the silhouette cast by an object that blocks a source of light. You can see your shadow on the ground or a wall or whatever you are blocking from the light source. You can make shadow puppets with your hands. A shadow can be cast by any light source such as a candle, a flashlight, an overhead light, a spotlight, or the sun.
Shade is the "darkness" created by a shadow and only really applies to shadows created outside by the sun. Underneath a tree on a sunny day would be "in the shade". Inside a building would be "out of the sun" and not "in the shade." However, under a roof with no walls (like a picnic shelter) would be "in the shade."
When used as a verb in the sense of "cast a shadow", shade and shadow can be used interchangeably. The trees shadow/shade the road.
When used as a noun, shade is relative darkness, shadow is what causes the darkness. A tree casts a shadow which causes shade. Shade is shadowiness. A shadow is shade within clear boundaries.
Historically they were the same word. In Old English, nouns were inflected, they had different forms depending on usage. The Old English noun sceadu (shade/shadow) became sceaduwe (in shade/shadow) when preceded by a preposition. Over time, the two word forms became interpreted as different words. The same process happened to the words mead and meadow. The uninflected form became archaic and is now used only poetically. (Mead in the sense of fermented honey has a different origin.)
So depending on context or idiom, they can't usually be used as a noun interchangeably. As well, each word has taken on other shades of meaning which aren't really equivalent. (Couldn't resist.)
What you probably want is
Let's sit under the shade
and
The cat was afraid of its own shadow.
A shadow of an object is formed on the ground when the object prevents the sun's rays from reaching the ground. A shade is an object used to shelter you from the sun.
However, shade is also used as an uncountable noun to refer to an area of darkness formed under a cover with the intention of keeping something cool, in which case one would use sit in the shade though sit under the shade is also possible.
Shade often deals with being sheltered from the sun and does not necessarily have strongly defined edges. A shadow may have more defined edges but aren't necessarily cast from the sun.
(BrE) Jasper Loy has made me realise that there might be some confusion here, as shade can have two meanings. There's the shade that Eric and ghoppe are talking about, which is sort of indefinite, for example from a group of trees, as opposed to the shadow from a single tree.
I think that is what would normally first come to mind for a native speaker. It's what you look for on a hot day. But here I would usually say sitting in the shade, not under the shade.
And then there's what Jasper is talking about, which we would normally call, I think, a sunshade. This is a bit like a large umbrella or parasol, and which you would indeed sit under.
@ghoppe - Oh, I do like shades of meaning - I've just been listening to 'Round the Horne' on the BBC. Do you know it? That joke would have been perfect for him.