Hi Emerellia welcome to the forum.
The most effective way to raise the conflict resolution is to use the friendly interaction 'Help Repair a Bad Relationship'. One way to do this is to have conversations with other NPC sims and hope that this interaction appears, there is a cool down on this interaction as well so once you have found a sim that you can use this interaction on you need to wait until it appears again.
The easiest way to manipulate this though is to have a vampire in your household and have them Compel for a Deep Drink on sims in a busy place. That way they will gain negative relationships without actually having to make them do mean interactions. Once a sim is in a negative relationship (takes a few drinks to get to the 'disliked' category of friendship and your vamp actually has to have spoken to the disliked sim at least once) then you can get your other sims to 'Help Repair a Bad Relationship' etc.
If you don't mind having one of your sims be a meany then make them be mean to another sim until they dislike each other which is a much quicker way to get the point your other sims can use the Help Repair a Bad Relationship interaction.
Apologizing after a negative interaction also raises the conflict resolution but it not as effective as using 'Help Repair a Bad Relationship' and also can lower the Manners Value in the process of being mean or mischievous etc in the first place.
Best thing of all to do is start this process when the target sim is still a child, by the time they are ready to age up into a teen they should be very close to gaining the trait. You can also complete the Social Butterfly Aspiration as they chat to all these random townies trying to find ones they can repair relationships with!