1 Know the rules
Here is a very brief explanation of the game's rules. Knowing the rules will help you win.
2 Save the game before playing
In order not to lose your important cards try to save the game first before challenging people. If you lost some important cards and couldn't get them back then just load the game.
3 Don't attack cards unless you have higher numbers
Don't attack a card that is Equivalent to you in power or even close. Make sure you attack other cards with significantly more powerful cards.
4 Avoid weak cards with many arrows
Don't use a weak card with multiple arrows. In such a case, the opponent can attack it with a more powerful card then let it flip all the cards it is pointing to.
5 Find the powerful cards
Many powerful cards are hidden during the game or can be achieved after completing some side quests. Make sure to have some strong cards. (See How to defeat Scarlet Hair in Final Fantasy 9?)
6 Take advantage of corners
Take advantage of corners by putting around them cards that can't be flipped easily or ones that don't have many arrows pointing to your cards (so that they don't get used against you).
7 Don't put a card with multiple arrows in the center
When you put a card like that in the center of the board the chances of that card being flipped through a fight might increase and as a result it might flip all the cards it's pointing to.
8 Flip powerful cards without battles
If the opponent has a powerful card then try to flip it by pointing to it on a side where it doesn't have arrows. This will help you avoid a fight and still flip the card.
9 Leave the powerful cards to the end
The powerful cards can help you win fights. You shouldn't start with them because you might not be able to save yourself at the end if things went wrong. (See How to defeat Ralvurahva in Final Fantasy 9?)