Difference between delighted with, about and at

It depends upon the object of your delight, although this is not a hard and fast rule.

You can be:

Delighted with your new purchase
Delighted for the success of your team
Delighted about the weather

In other words:
Delighted with is usually used when referring to an object that pleases you.

Delighted for (or delighted at) is usually used when referring to something done by someone else – it is their happiness that pleases you.

Delighted about is usually used when referring to a situation that pleases you, especially one over which you have no control.

However, you will see these forms used interchangeably. It would be wrong to become too strict about their use.