Sure, I can give you some tips here and, if you'd like something more in-depth, you can you email me at firstname.lastname@example.org and I can give you a full rundown.
So, it can depend on who you are trying to romance, but basically following your mother's initial advice and behaving like a frivolous twit is what is going to earn you the King's favour.
Off the top of my head, a few of the 'easy wins' are being all shy when you first greet Seneschal Corin, not eating at the first ball, playing the Rose Princess in the play and playing her to the script, crying when you have the opportunity. Fainting is also always a great boost for your points, lol.
There are many more points available than are required to have the King choose you at the end, so you don't have to do everything right. Also Bastien mentions on Day2 and Day7 how well you are (or aren't) doing with regards to the King's favour, so that should give an indication of whether you're on the right track.
You'll know if you've managed it at the end because the King will ask your MC to step forward not Francesca - of course, whether you're there to hear him is up to you. ^_^
Thanks for playing