What is the religious ruling on a husband who discovered that his wife is cheating on him with another woman, and what is the correct course of action he should take?

If it is proven that your wife has betrayed you through lesbianism or the like, then advise her and remind her of Allah and the evil of her deed. Try to reform her by nurturing her in religion and faith, connecting her with Quranic memorization centers and circles of knowledge, and encouraging her to associate with righteous women. Increase your supplications for her. Do not resort to divorce unless its benefit outweighs its drawbacks, in order to avoid its negative consequences, especially if there are children.