The rules of courtship change, and as with everything in life, different rules apply for different personalities. Here is a rough and ready primer that takes you through a few do's and don't' s (they are both for him and her.
DO'S
For her
Tell him you adore his muscles
Roll your eyes in disgust whenever the subject of your ex-boyfriend comes up
Remember to compliment his mother on her culinary skills
Pay attention to his little sister's hobbies and heartbreaks
Offer to pay occasionally on a date if you're both earning comparable salaries. If you insist on going dutch all the time, more power to you.
Compliment his taste in clothes/food/décor/music/art. Whatever applies.
Tell him you love him. Often.
For him
Tell her she looks beautiful
Refrain from making hooting noises when a bombshell walks past the two of you.
Hold her hand at the most schmaltzy moment in a romantic film.
Tolerate, even sing along to `your song'.
Avoid all comparisons with your ex's.
Burn your black book.
Be a gentleman. Pick her up and drop her to her home on a date. Open the door for her. Guide her through crowds. It is always appreciated.
Call after a wonderful evening and tell her it was wonderful.
Call when you say you will call.
Send her flowers.
DONT'S
For her
Don't let him ever see you with your green cucumber face mask
Don't Pee with the door open
Don't discuss your menstrual cycle
Don't try to make him jealous by flirting with the stud at the bar
Don't refer to his mother as that woman
Don't forget the deodorant
Don't buy him gifts that are too expensive until you are exclusively a couple.
For him
Don't belch, fart, nosepick or do any unmentionable scratching in front of her
Don't rush to grab a seat in the theatre before she does.
Don't order for her from the menu
Don't expect to be forgiven for everything.
Don't forget any anniversary/birthday or momentous occasion
Don't fail to comment on a change of hairstyle or a new outfit
Don't forget the names of her best friends/siblings/great aunt/ godmother
Don't expect her to sink into your bed just because you paid for the dinner and the disc
Don't buy her personal gifts like lingerie or clothes until you are sure of her tastes. Or until you know her sizes right.
Don't buy utility items as gifts. A toaster is not romantic. Nor is a handblender.