Not only X, but also Y
The word “do” will be very useful here. If you remember the three forms that the present tense can take — “I run, I am running, I do run” — then you’ll recognize the proper use of “do“.
Not only do institutes provide X, they provide Y.
Not only does the hotel provide X, it provides Y.
Not only does the hotel provide X, it also provides Y.
Someone who wants to make it a simpler sentence might say:
The hotel does not provide only Wi-Fi, it also provides breakfast.
The hotel does not only provide Wi-Fi, it also provides breakfast.
The hotel provides not only Wi-Fi, it also provides breakfast.
The hotel does provide both breakfast and Wi-Fi.
The hotel provides both breakfast and Wi-Fi.
Parts of speech
When you use "not only X but [also] Y", X and Y should be the same part of speech (type of word). This is called "parallelism".
It can be two verbs:
Not only does the hotel provide breakfast, it washes your car.
It can be two nouns:
The hotel provides not only apples, but [also] bananas.
The institute provides not only training, but also support.
The institute does provide not only training, but also support.
It can be two adjectives:
The hotel is not only historic, but [also] beautiful.
When the subject is different
In your example sentence you wrote a sentence like this:
Not only do institutes provide XXX, the elderly receive YYY.
This is grammatically okay: both the first clause and the second clause look like "verb noun".
