Which is the best fast-growing tree to plant for shade?

Maple

I'm not sure how fast growing it is, but I love Aspens! Plus they sound good when the wind blows.

If you have wet soil you should try a birch or willow tree.

It depends where you live. Varieties of maples grow all over the United States and you’ll need to check with your local county extension office on what they recommend.

Keep in mind that the faster a tree grows, the more brittle it is.

This means in a big storm, any time of year, fast growing trees are also highly susceptible to severe limb damage.

That’s fine if you don’t care about the health of the tree in the very long term, or you don’t care if a limb blows onto your roof or your car.

Cottonwood. They get big very fast. They can be very messy however with losing their branches. What you plant for shade trees will depend on where you are planting them. For example, you don't want cottonwoods next to your house but they work great planted in a row along the edge of a large yard to block west sun and shade your yard from the hot afternoon sun. What you plant will also depend on where you live. Not all trees do well in every climate. You may consider visiting a local nursery to see what they suggest for trees in your area.