What are the legitimate Islamic ways to search for a wife?

You must take the initiative to marry, in obedience to Allah and in compliance with the command of the Prophet, peace and blessings be upon him, and as a means of safeguarding yourself, and seeking righteous offspring. Begin by performing Istikhara (prayer for guidance) and seeking Allah's help, then consult those whom you trust for their piety and intellect. After that, search for a suitable wife, and make religion and character your priority in selection, for the Prophet, peace and blessings be upon him, said: "So win the one who is religious, and you will prosper." Do not neglect other criteria such as beauty.

You can ask the righteous women among your relatives to help you find a suitable wife. If options are presented to you, perform Istikhara and choose one of them. Then, try to meet her and look at her, for the Prophet, peace and blessings be upon him, said: "If one of you proposes to a woman and is able to see some of what would encourage him to marry her, let him do so." The majority of scholars do not require the permission of the woman or her guardians for this viewing. You may look at her face, hands, and stature, and speak with her to discern her intellect and reasoning.

If you find what pleases you, propose to her and seek Allah's help. Do not seek absolute perfection. Ensure that your meeting with her is in the presence of one of her mahrams (unmarriageable kin) or one of your female mahrams. After the meeting, refrain from any contact with her until the marriage contract is concluded, because she is still a non-mahram to you. After marriage, treat her well and fear Allah regarding her; for she is a trust with you.