Answer: Is there a correct way to go about dating according to the Bible? Do you make a shopping list, and check off each step? Did this. Check! Did that. Check! Trying to find a mate this way would be as exciting as watching paint dry.
Courtship is like the rest of life. It has many twists and turns, and you deal with them as they come. God's word says very little about dating because wives were generally chosen by the parents.
When I was a teenager (more than fifty years ago), no one picked a girl and decided they would be dating. We went out with many girls, with of course no kissing or sex, just a good time at the beach, or bowling, or at the movies. Eventually the chemistry of love began working between a particular couple, and it became apparent they were meant for each other. Only then did they begin see each other on a regular basis in what we refer to as dating today.
In my own case, I found myself deeply in love with a very beautiful girl, and we began thinking about marriage. However, she went to college where she met a fellow, and wrote to say they were going to be married. That hurt, but God had better plans.
One of the girls I had been dating was waiting in the wings. When the letter came from college, Jeannette made sure she was at the head of the line for dates. At age 21 we married. We had a wonderful life together for three years, until she got sick and sadly died at the age of 24.
Life then took me far from home, where I met another girl. After a short span of dating Betty and I married, and have shared life happily together for more than forty years.
The nearest thing we have to dating (courtship) in Scripture is in Genesis 28. It is the touching story where Isaac told Jacob to go to Padan Aram and marry one of his uncle Laban's daughters (Genesis 28:1 - 2).
The narrative jumps to Genesis 29 where Jacob falls head over heels in love with his cousin Rachel. He then declares his willingness to work as his uncle's hired hand for seven years for Rachel. When his uncle tricked him into marrying Rachel's older sister, Leah, Jacob agreed to work another seven years to have Rachel. His love and devotion to her is truly amazing (see Genesis 29:1, 4 - 5, 9 - 12, 16 - 18, 20 - 22, 25 - 28)!
One little bit of advice regarding dating. Do not make the mistake that sex is the same as love. If the person you are going out with wants you to prove your love by letting him use your body, there is not much love there. Prove your love, and let him prove his, by saying no to sex during any dating period until you are married.