The fighting-with-spouse aspect of childrearing is not inevitable. These are inevitable for parents:
You will be tired.
You will be faced with things you have no idea how to handle, fix, solve.
You will not agree with your co-parent on every detail.
You will disappoint each other.
You will have bad moods, and they often will be exacerbated by lack of sleep, especially early on.
You will have responsibilities that are boring, repetitive, relentless, mildly irritating, of great consequence to your bond with your child, and rewarding mostly in the very long term. (Diapers, feedings, play ...)
Obviously, it's easy to see why the combination of these on a daily basis over years can lead people to fight. But, again, it doesn't have to. Probably the biggest thing you both can do to prevent bickering is to put everything you've got into this. That means not rolling over and making it the other person's turn to get up unless it actually is. It means seeing what needs to be done and doing it, instead of hoping the elves will get it. It means communicating--"I do plan to do the dishes, I just need to sit for a second."
When both of you can plainly see that neither of you is taking advantage of the other, then you can use these other, highly effective fight-preempters:
Recognizing the other person is tired, too.
Occasionally giving the other person a break, even when it's technically not your turn.
Admitting when you're faced with something you don't know how to solve, and asking for ideas.
Leaving it alone when you disagree on something small--it's okay for kids to have two different experiences with two different parents--and taking it sit-down-and-talk seriously when you disagree on something big. It's not okay for there to be such great differences that kids (at surprisingly young ages) know they can use you against each other for leverage. It's also not okay when one parent insists on something genuinely risky (texting while driving, say).
Apologizing when you let your spouse down, and forgiving when he lets you down.
Finding family-workable ways to deal with your bad moods. Will a walk do it? A trip to the gym? A loosening of the rules that day? (Ice cream for breakfast once in a while will not derail the train.) Make sure you both have outlets the other knows about and is willing to provide when you cry uncle.
Last but not least, build something into your schedules that serves to remind you of why you like each other. Date night, a favorite show, a favorite team, etc.
I'm sure I missed a bunch, but by now you probably all think I've fallen off my chair.