How to change signature officially on Passport.

Yes, it is very much necessary to update your new signature in your passport. The procedure is almost similar to the process of a new passport application.
You can do it online as well as offline. In both cases, you need to apply for the update in personal details. There you need to tick/ mention the signature box. At the time of your call in the passport office, they will verify your old signature and then will match your new signature with other legal documents. Once all these documents are verified, the address verification can or cannot be initiated depending upon the verification of your documents.
If the police verification is initiated, you will receive an SMS regarding the same. In other cases, you will receive your updated passport within 15 days. In either of the case, you can check or track your application status online.

If you want to update your signature or if you want to change your signature you have to follow the procedure as if you are applying for a new passport. The applicant has to append his old signature and new
signature on the second page of the passport application form. It can be done offline or online also.
When you apply for this the following documents are to be attached to your application. Before sending please ensure that you have filled all the columns of the application form correctly and completely. Please don't forget to sign on the application where ever required.
1. 4 35 mm x 45 mm size passport size identical photographs. The photo should be of the white background. The photo has to show a frontal view of full face.
2. The current passport having a valid visa or residence permit is required to be enclosed.
3. If you have additional booklets, those are also to be attached.
Then the normal course will follow and you will get the passport. If you want it urgently there is a Tatkal process also.

" Please don't forget to sign on the application where ever required." - so I need to use which signature for this- old signature in passport which I want to change or new signature which I want to use hereafter?

You have to use your new signature in all the places when you are applying for a change of signature. Only in one place, you have to append your old signature. This is very important. You are not supposed to use your old signature anywhere.

The changing of signature can be done for the passport and you need to follow the same procedure as you do when one is applying for a new passport. You need to make a fresh application with amendment i.e. new signature on the form and the old signature should be on the second page of the application form. You need to provide the following documents along with your application form:
1. Four (4) passport sized identical photo with full front view of face with white background.
2. Submit the current passport and if it has additional booklet, do submit the same along with the passport.
3. Duly filled application form with signature.

You can have a view of the attachment for reference.