Praise be to Allaah.
Islam came to protect women and their honour, and prescribed rulings that take care of that. Allaah says (interpretation of the meaning):
And stay in your houses [al-Ahzaab 33:33]
Based on this, the general principle is that women should stay at home and not go out except for essential reasons or cases of need. Islam states that a womans prayer at home is better for her than her prayer in the mosque even al-Masjid al-Haraam.
This does not mean that women have to remain prisoners in the house. Islam permits them to go to the mosque, and has made Hajj and Umrah, Eid prayers, etc. obligatory for them. Among the kinds of going out that are prescribed are her going to visit her family and mahrams, and going out to ask religious questions from people of knowledge. Women are also permitted to go out for their own needs, but all of this has to be within the limits and conditions set out in shareeah, such as being accompanied by a mahram when she travels, or being assured that her route is safe when she moves about in her own city or locality. She should also go out wearing complete hijaab, and she should not be wearing make-up, adornments or perfume.
A number of texts have been narrated concerning this, including the following:
Ibn Umar reported that the Prophet (peace and blessings of Allaah be upon him) said: If the wife of any one of you asks for permission to go to the mosque, do not stop her.
Zaynab, the wife of Abd-Allaah, said: The Messenger of Allaah (peace and blessings of Allaah be upon him) said to us: If any one of you (women) comes to the mosque, let her not wear perfume.
Jaabir ibn Abd-Allaah said: my maternal aunt got divorced and wanted to go and pick some fruit from her trees. A man told her off for going out, so she went to the Prophet (peace and blessings of Allaah be upon him), and he said, Never mind, go and pick the fruit from your trees. Maybe you will be able to give it in charity or do something good with it.
The entertainment referred to in the question may involve mixing or looking at strangers (non-mahram men), or travelling without a mahram, or many things that are of no benefit. So you have to be cautious and make sure that the entertainment really is permissible and halaal, and free of any haraam things that would earn the punishment of Allaah. If a woman goes out to a place where there is nothing haraam going on and she does not go out too frequently, there is nothing wrong with this. We ask Allaah to protect us, keep us chaste and make our commitment to religion good and strong. May Allaah bless our Prophet Muhammad,
Sheikh Muhammed Salih Al-Munajjid