Are Women Allowed to Go out in Islam?
Question

I've been wondering about what I've heard some Muslims say about women going out, that they have to have some legitimate purpose to go out. Would going out for things that serve little purpose (like halal entertainment of some sort) as long as the woman wore the proper Hijab? I and another Muslim girl differ on this.
Praise be to Allah.Purpose of Islamic guidelines for women
Islam came to protect women and their honour, and prescribed rulings that take care of that. Allah says (interpretation of the meaning): {And stay in your houses.} [Al-Ahzab 33:33]
The general principle of women's stay at home
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 woman’s prayer at home is better for her than her prayer in the mosque, even Al-Masjid Al-Haram.
Are women allowed to go out in Islam?
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. 
Conditions that govern women’s going out
Women are also permitted to go out for their own needs , but it must be within the limits and conditions set out in Shari`ah, 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 a complete Hijab, and she should not be wearing make-up, adornments or perfume .
Narrated texts on women's mobility
A number of texts have been narrated concerning this, including the following:

Ibn `Umar reported that the Prophet (peace and blessings of Allah be upon him) said: If the wife of any one of you asks for permission to go to the mosque, do not stop her. (Narrated by Al-Bukhari, 827 and Muslim, 442)
Zaynab, the wife of `Abdullah, said: The Messenger of Allah (peace and blessings of Allah be upon him) said to us: If any one of you (women) comes to the mosque, let her not wear perfume. (Narrated by Muslim, 443)
Jabir ibn `Abdullah 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 Allah 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. (Narrated by Muslim, 1483)

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 free of any prohibited things that would earn the punishment of Allah. If a woman goes out to a place where there is nothing prohibited going on and she does not go out too frequently, there is nothing wrong with this.
We ask Allah to protect us, keep us chaste and make our commitment to religion good and strong. May Allah bless our Prophet Muhammad,