Some people reason that dating a non-Christian is no big deal since they hope to eventually convert the unbeliever. But dating in hopes that you will change someone is taking a great risk. Since you will likely end up married to a person you date, I will advise my own children to date only Christians.
While some unbelieving spouses are converted by their mate, this is the rare exception to the rule. So the book by Lee and Leslie Strobel, Surviving a Spiritual Mismatch in Marriage, adamantly warns against "missionary dating."
Suppose you're engaged or married to an unbeliever. He may finally get baptized or join a church in order to foster a "united" marriage. But you may always be well ahead of him spiritually. He may never read his Bible outside of church, pray with you outside of meal times, or grow as a Christian spouse.
You may date someone who claims to be a Christian. He may attend church with you. But is he really a committed follower of Jesus? He may attend church to impress your family, or to win your hand in marriage, or just to be with you. But once you say "I do" his commitment may degenerate to anywhere from marginal to non-existent. I've seen this happen to couples on countless occasions.
Some might justify dating non-Christians, insisting that they are nicer, or even morally superior to the Christian dating options. That will only be true when those claiming to be Christians are hypocrites. A true Christian will strive to follow Christ in all things.
Some might protest that there are few eligible Christians in their small town or small church. So attend a Christian university or move to a different city when you're old enough to be on your own. Better not to date at all then to put your soul in harm's way by dating a non-Christian.
Marrying a non-Christian could mean a lifetime of disagreements on everything from debt, to drinking, to entertainment choices, to whether to give to the church. The biggest disagreement will be child-rearing.
I once heard a statistic that if the father in a house is not a Christian there is a 75% greater likelihood that his children will not be Christians. Many have beat these odds, but the percentage sounds reasonable to me. If only one parent considers something to be important, the children aren't as likely to think it's important. The bottom line is that a non-Christian influence in the home could actually mean that your children run a greater risk of going to hell. That may sound harsh, but life will be much easier if both parents are on the same spiritual page.
I would personally not even recommend marriage outside one's own fellowship. A Catholic/Protestant or Evangelical/Liberal mix could bring a multitude of problems if either party is truly dedicated to their tradition. The big question is where the kids will go to church. If religion is important to you at all, you'll want them to go with you. To be a truly committed disciple, while married to someone who is not is a recipe for frustration and possibly disaster.