I am the joint holder in both the shares as well as the FDs.
Assuming you were the joint holder only for convenience and the money actually belongs to you both. Even otherwise it does not change much.
For Shares the best way is have your sister open an Demat account. Transfer half the share to her via offline transaction. As the amount is large for an offline transaction, Bank or Income Tax may ask for clarification.
Option 1: You can submit your father's death certificate and an affidavit stating that you both jointly agree/decide to split these in the ratio of 50%.
Option 2: The other option is you execute a gift deed and transfer shares to your sister. Gift between close relations is tax free.
So either ways there will be no tax implications for you or your sister on this transaction. If you sister were to sell these shares, the date of vesting in option 1 would be when your father expire. i.e. if the value of share at that date were say 100 [doesn't matter at what rate your father purchased], and she sells for 150, 50 becomes the profit and depending on long/short term taxes will be applicable. If you go with option 2, the date when you execute the gift deed will be vesting date to your sister.
As you were join holder of the shares, I am not sure what would be considered a vesting date for you. Consult a CA who will be able to advise you better for a nominal fee of Rs 2000 - 3000.
Similarly for FD's you can transfer the ownership to your sister or liquidate them and transfer the cash.