Have you thought about the fact that there are some topics that are inherently intimate and that almost automatically promote deep intimacy between two people? What do I mean?
For starters, let me suggest that you not go out in the first week and tell each other the long, teary versions of your testimonies and the greatest personal pain that the Lord has delivered you from in your life.
Don't immediately make that person your confidante in matters personal and emotional. Don't articulate your deepest feelings with respect to your life or even how you feel about that person. Also (and this may seem counterintuitive), I advise folks not to spend long periods in prayer together. Prayer is a wonderful thing, but it's also inherently intimate. Pray for the relationship, but don't spend hours holding hands and pouring yourselves out before the Throne. That may come.
What should you talk about then? Talk about a book you're reading, your interests, your faith (in more general terms or along the lines of issues), things going on in your life. Talk about your values and priorities, ambitions and plans you may have, your families and things that are happening in your church or in the world.
All right. Does this sound cold, uninviting, even deceptive? I admit it's not the stuff of movies, but the very point that I'm making is that at this point it shouldn't be. You are not yet that other person's main provision from the Lord for spiritual, emotional and physical intimacy and companionship. That role is reserved for the person's spouse. You are not that yet. You are in the early stages of seeing if that is a role that the Lord would eventually have you fill in one another's lives, but you're not there yet, and the kind of intimacy I've described is not to be engaged in on a trial basis. Even if it looks more fun or stimulating to go there — and I know it does — it's also defrauding your brother or sister.
This brings me to the larger principle bound up in these suggestions: Deep emotional intimacy should not be established in the early stages of a relationship.
It's not that you're being dishonest or cold, it's simply being cautious about living out a deeper commitment than truly exists between you. Song of Songs 2:7 tells us not to awaken love before it pleases: Do not start what you cannot — without sin — finish.
The modern, secular idea of dating relationships is to test the waters of marriage by acting as much like you are married as possible until you both (in the very heat of that temporary emotion and passion) decide what you want and either get married, or until one of you decides it's not a good fit and you go through something like a divorce (at least emotionally, if not physically — though that's pretty common, too).
The biblical idea of marriage holds that such level of relating to one another begins when you are married. It's one of the things that makes marriage unique. Our goal should be prayerfully to decide whether the person we are dating should be the one we marry without having to go through a de facto divorce if the answer's no.
Will there still be disappointment and sadness and emotional pain if a "biblical" dating relationship doesn't work out? Of course. There's no perfect way to do this. I assure you, though, that the pain will be lessened by the honest, mutual, spiritual concern for one another that results when two people treat one another like brothers and sisters in Christ first, and potential spouses second. This is for the protection of the people involved (especially the woman), for the witness of the church and for the glory of God.