This has been tricky for me to understand the difference in meaning between them, so which one is wrong or doesn't sound appropriate?
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.
Addressing a specific question from the comments:
What would be correct
I am delighted at being offered an admission in...
I am delighted to be offered an admission in...
Both are certainly correct.
The former (delighted at being) assumes you have already been offered admission, whereas the latter (delighted to be) is slightly -- but only very slightly -- forward looking, with an implied "if... it actually happens". The fact that you have said "am delighted" already indicates that you understand you have been offered admission. If you had said, "I would be delighted to be offered..." it is clearly forward looking, and there the "at being" would be incorrect.