Jana, based on your name (and your picture), I assume you’re probably from a Slavic country. It’s important to realize that in English we are not as sensitive to the level of formality used in a text as people are in Russia, Poland, the Czech Republic, Slovakia, etc. Also, the level of formality that Slavic speakers would use in a business report or job interview would be quite outrageous in English. One of my friends failed a translator certification exam partly because he retained the formality of the Russian original in his English translation. The level of formality in a typical Slovak business letter, for example, would be quite wrong for the English-speaking world.
As for when to use contractions and phrasal verbs, the situation is not as clear as “use them or don’t use them”. Some contractions and phrasal verbs are acceptable in formal language and some are not.
Contractions
Contractions like gonna, wanna, ain’t, woulda, coulda, shoulda are usually not acceptable in writing. Generally, they’re used in comics or other texts at that level.
Contractions like it’ll, would’ve, could’ve, might’ve, that’ll can be used in informal writing to friends, but they’re not usually found in a newspaper, for example.
Contractions like won’t, can’t, don’t, doesn’t can be used in almost any text, even formal ones, if they sound right. Remember that sometimes the full forms, such as will not, cannot, do not, does not, may sound arrogant or angry, and sometimes they are too emphatic.
Phrasal verbs
There are a lot of phrasal verbs that are perfectly fine in formal writing. One of them would be take up, which can sound quite formal. We can say, “We will now take up the matter of next year’s budget,” or even, “He took up his knife and defended himself.” Both of those sound formal. Very many other phrasal verbs are acceptable in formal writing.
Keep in mind also that sometimes we have to use a certain phrasal verb because there is no other word that expresses the right idea. Another matter is that in order to replace a phrasal verb, the only other alternative may be a long word from Latin or Greek. Any English – formal or informal – that contains no phrasal verbs at all, and replaces all of them with Latinisms, will sound comical to native speakers.
There’s no real rule about when you can or can’t use contractions or phrasal verbs. The only way to understand when to use them is to look at a lot of writing at different levels of formality.