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! Try 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.
The Bible says very little about dating because wives were generally chosen by the parents. The nearest we have to courtship begins in Genesis 28:1 - 2 where Isaac told Jacob to go to Padanaram and marry one of his uncle Laban's daughters.
The narrative jumps to Genesis 29 where Jacob fell head over heels in love with his cousin Rachel. Jacob declared his willingness to work as his uncle's hired hand for seven years for Rachel. When his uncle tricked Jacob into marrying Rachel's older sister, Leah, Jacob agreed to work another seven years to have Rachel. What devotion and love he had (Genesis 29:1, 4 - 5, 9 - 12, 16 - 18, 20 - 22, 25 - 28)!
When I was a teenager (fifty-something 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.
Even then, however, everything didn't necessarily go smoothly. 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 often dated, a short, real plain Jane, had been waiting. 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 courtship Betty and I married, and have shared life happily together for nearly forty years.
One last little bit of advice I can offer regarding dating is to NOT make the mistake that sex is the same as love. If he wants you to prove your love by letting him use your body, there's not much love there. Prove your love, and let him prove his, by saying no until marriage. Thank you for your question about what they bible says about this important topic.