What are the conditions which are needed to be fulfilled before marrying a second, third or fourth woman?
Edit
Apart from the above part the thing I am actually looking is something that says that the current wife / wives permission is needed to be taken for the next marriage.
Before marriage, financial ability and not having more than 4 wives is enough but Islam does not allow any discrimination between the wives or between their children. The Holy Qur'an expressly says:
If you fear that you will not do justice (to them) then have one only (Surahan-Nisa,4:3)
Reference:
You said that you are looking for something that forces current wife/wives permission for next marriage. Actually, in contrary with what you are looking for, Allah forbids banning a halal to gain consent of one's wives:
يَا أَيُّهَا النَّبِيُّ لِمَ تُحَرِّمُ مَا أَحَلَّ اللَّهُ لَكَ تَبْتَغِي مَرْضَاتَ أَزْوَاجِكَ وَاللَّهُ غَفُورٌ رَّحِيمٌ ﴿التحريم: ١﴾
O Prophet, why forbiddest thou what Allah has made halal to thee, seeking the consent of thy wives? And Allah is All-forgiving, All-compassionate. [66:1]
So according to Qur'an (Nisa 4:3) not discriminating is enough,
If a man has the ability to take one more then one then it is his right, but if he cannot then he should marry only one:
فَإِنۡ خِفۡتُمۡ أَلَّا تَعۡدِلُواْ فَوَٲحِدَةً
but if you fear that you will not be able to deal with them justly then marry only one
Furthermore a man does not have to seek permission from the current wife to take another. For part of the wisdom of taking another wife is to help the society in whole, for it is natural that women (in most societies) are more then the men, hence taking more then one (wife), can ensure that the society stays balanced and that women can find husbands, and Allah knows the full wisdom. Furthermore it would be natural for a women to be jealous and not want to share their husband, but this jealousy should not prevent the man from what is ok/halal/permissible for him to do. So basically to get back to the point, it is not a prerequisite nor an obligation to seek a wife's permission to marry another.
Main Sources: Islamic online university
Though The QURAN permits polygamy for men (4:3), it severely discourages its actual practice by requiring certain significant preconditions: men may marry more than one wife only if the later ones are widows with children, and they should treat each wife equally and fairly. (See 4:19-20; 127-129.).
