There's nothing wrong with arguing in a relationship, its common, not everyone will agree on all the same things. however, look at the things you are arguing about. You said you both fight about the small things; honestly I've learned that somethings you just have to let go. Don't let your emotion run high on such a small thing like who's turn it is to throw out the trash etc.. some things can be avoided. Relationships can be hard, and you may not always see eye to eye, but try not to blame each other during these fight, instead express your feelings, and find constructive ways to solve the issues. But remember pick your battles. If you guys both care about each other, you guys will make it work. You just have to learn and understand your partner and their difference and have constructive arguments that better the relationship, not ruin it.
I used to struggle with this problem too. I would pick out fights out of thin air over small or big issues. He would be speechless, not knowing what to say and I would lash out names at him. We go to bed angry and refuse to solve the issue the next morning. The cycle continues.What you need to do first, is build/fix communication. Second is get on the same page with how you deal with issues. For example, your partner likes to lash out while you prefer to deal with a cold-head. Get him to calm down and then try to resolve the issue. Third, promise to not fight over the same things because it is unnecessary and it's only going to hurt your relationship.Another thing you need to bear in mind is that there are not only YOU and HIM. There is also this RELATIONSHIP. Choose to do the things that are not going to hurt the RELATIONSHIP.