What is the best time for Salatul-Istikhara, and is it true that after the prayer, one should not speak?

The best time for Istikhara prayer is the last third of the night, because it is a time when supplications are answered. However, it is permissible at any time except during the forbidden times. As for not speaking after it, there is no original basis for this from the Prophet, peace and blessings be upon him.