Or should you spell it out? Like 'it is' instead of 'it's.' I was just reading somewhere where they said that grammar teachers would prefer if you spelled it out. Maybe it was the other way around, but I was wondering what is the more proper way to write? With contractions or without contractions?
there's no fixed rule... some teachers will accept contractions in some cases, some won't... it's not a matter of what's 'proper' but only of what's called for in each instance... in fiction dialog it's expected... in formal essays, it's often frowned upon... and so on...
They're both correct, but in different contexts. Spelled out is formal or emphatic, contracted is informal. And there are places you can't contract, of course: you can't (in normal speech and writing) contradict a negative statement with "Yes it's!" You have to expand it to "Yes it is!". Grammar assignments are formal things so spelled out would normally be the expected thing -- the same is true of pretty much all academic writing. Dialog is usually informal, so contracted is usual unless the character is making a point or is a very formal sort of person. For narrative it depends on the sort of narrative voice you want to have.
What is the voice you're aiming for? Do what sounds right for that voice. Back when I started writing, I never used contractions, because I was going for a kind of formal, almost archaic voice, something that wouldn't have felt out-of-place in a seventeenth-century manuscript. More recently I've loosened up a lot, and I find that using contractions can often increase the poetic feel of the voice. In musical terms, I stopped trying to sing opera and started singing folk songs. It works better for me.
Since people have already stated the proper/improper uses of contractions, I personally find that it depends on the character. (With the narrator I usually don't contract, where as with a laid back character I usually do.) And it does depend on the voice you're trying to have. That would be an important consideration.
In dialogue, I feel free to use contractions. In prose, I do not, unless it sounds really robotic without it. Regardless of formality, that's what you really have to look at. Expanding your words can often sound robotic; it lacks emotion and such. This is one of the "problems" (it's not really a problem since it's done on purpose) with Faye's early dialogue in Questionable Content (look it up (NSFW?)). She sounds kind of robotic, and other characters mention this, semi-frequently.
I feel that the premise of the story kind of relates to whether you use contractions. For example, writing from the perspective of a oh so upper class lady wouldn't make sense if you started adding don't, can't, won't or, heaven forbid, 'init'. Then again writing from the perspective of a child or just a normal person makes sense to use a few contractions in prose, if the 'proper' word or phrase sounds weird and doesn't flow properly.