How to get them to start liking each other as friends, let alone 'like like'

So I'm a fantasy writer, but I love romance. I have my main character, female, who's a mage. Trossie is a fire mage, but she's cold, distrustful, and aloof. The love interest, Drey, is...sort of the same? He's good with multiple weapons but used to archery and swords; he's distrustful of magic and those who use it. He was assigned to my MC as a guard when she got kicked out of the capital for having a tantrum, and he distrusts her specifically for that. He also thinks she's basically a child in an adult body, and he sort of looks down on her.

How to I get them together? I'm having trouble with the awkward I-dislike-you-but-I'm-disliking-you-less stage and getting them to ease from detesting each other to being sort of into each other. And I'm really bad at movements in character like that. The whole change in feelings is meant to take place while she gets better with people and learns to not be a stuck-up horrible brat.

Any suggestions on making it work and keeping it realistic? I can't stand when characters turn around one day and love each other after a whole novel of dislike, and could use some pointers on avoiding that.

I love doing this sort of things with characters!

Have them get into a situation where they have to depend on each others skills. Have them bond over something, maybe she sees someone down on their luck and decides to help them and he sees her in a different light because of that. Maybe he is a "I can pick at her but if you screw with her, I will break your neck" type. Have someone get the upper hand on her and be a total prick about it so he can step in.

It doesn't have to be a huge "OMG I AM IN LOVE NOW" Just a "Huh....you are 2% not douche" I actually wrote a scene where one character saves his future love interest simply because he didn't wanna hear someone complain if she died and he finds her slightly useful to their cause. He tells her that to her face in front of everyone. Needless to say, they do not fall in love anytime soon after that, but that double handed compliment does start a series of debate sparks between them. They even bond over their mutual hatred of someone. XDD You could always go that route, when in doubt....fight it out.

I, personally, love it when two character's hate each other so much, but then they go at it. And it's so passionate, because they can't stand each other's personalities but are so attracted to one another. And then they slowly start (unintentionally) opening up to each other, and each one sees the other differently with each passing day. I love when relationships still have that fiery-ness (is that a word?) and it's not all sweet and loving and tentative, but yet they've got each other's backs 100%.

Either that, or write in a romantic interest who is "more his/her type" and have the other get jealous - but not know why. ;)

So many different things that can be done... Try a few, and see which one fits. I have a feeling that they'll "tell" you when you've it hit, though!