Is there a correct way to go about dating according to the Bible? Unfortunately, attempting to find someone you might be compatible with is more complex that simply checking off boxes on a form. Looking for a potential mate takes far more work and time than creating a list of stuff as if you would for grocery shopping.
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 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.
Happiness and sorrow
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. We married at the age of 21. 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.
A Biblical example
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)!
A little advice
One last little bit of advice. 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. This is a principle fully backed by the Bible.