Praise be to Allaah.
The Prophet (peace and blessings of Allaah be upon him) said: “If there comes to you one with whose religious commitment and character you are pleased, then marry (your female relative) to him.” And he said, “Seek the one who is religious.” What is addressed to men includes women too.
When you find a young man whose religious commitment, character and attitude you are pleased with, then you have to accept him as a husband. What will help a girl to do that is to ask about this man from those who know him well, because during short visits, and especially when one is seeking to get married, people are usually on their best behaviour and one sees little of a person’s true nature and character.
Shaykh Muhammad al-Duwaysh.
A girl may also feel somewhat afraid when thinking of marriage to a person because it is such a life-changing matter. But this fear should not stop you from agreeing if he is religious and of good character.
We would also draw your attention to the issue of your sitting with this person, which you mentioned at the beginning of your question. If the betrothed couple sit together in the presence of someone who will ensure that there is no khulwah [being alone together], and the woman wears hijaab as she wears for prayer (i.e., covering everything except the hands and face), for a length of time which is sufficient for both parties to make a decision, this is correct and is what is prescribed in Islam; if it is anything other than that, then beware…