It really depends why you are learning.I am not sure what you mean by "street language" because even if you want to just talk to people,you still need to know how to talk correctly,not just in slang.But if you mean "Do I need to learn grammar?" I think you need some because otherwise nobody will understand what you are saying,if you make many mistakes.Maybe you don't need to learn a lot of grammar rules,just the basic ones.

One main difference between Arabic and English is that different Arabic-speaking countries have different dialects and you need to switch to standard Arabic to understand each other. This is not the case with English. There are regional differences in English, but these are all very minor. Native speakers over-exaggerate how big regional differences are. These differences are almost entirely restricted to accent and slang and they more likely to just sound strange to others than they are to cause misunderstanding. A person from Texas might think a person with a Cockney accent sounds bizarre and thus might insist that their English is completely different, but they shouldn't have any problems understanding each other. English doesn't really have true dialects in the same way as Arabic. This may not hold true when you look at the language spoken in certain parts of Scotland, but there are arguments about whether the language there spoken is a dialect of English or a related but separate language. The "standard" English you'd be taught from textbooks allows you to understand 100% of any so-called street language you might hear in any English speaking country and if worst comes to worst, they can just ask what the other person meant.

Part 2: If you put a group of people from London, Sydney, New York, Toronto and Glasgow in the same room, they'd all be able to understand each other without any problems speaking the same way they would speak to their parents. In fact that's what native speakers do do. Native speakers from difference regions don't switch to "standard English" because there's no such thing: there's no need. The main difference you need to worry about is accents. Australian, Texas, Boston, Cockney etc. accents all sound different, but not to the point where people can't understand each other. People might mishear some words, but if they do they can just say "pardon". English speakers don't consider any accent more standard than any other. Vocabulary can also differ, but in my opinion the differences in vocabulary between regions are smaller than the differences in slang between generations. Some phrases might also be different (ex. an American might think saying "How are you going?" instead of "How are you doing?"), but this rarely causes major misunderstandings. Whereas people on Arabic TV might speak standard Arabic, producers of movies and television shows here love to give different characters completely accents. As such, native speakers are often familiar with slang from other regions. I can't think of a single movie where all characters all spoke just one type of English. So is the English grammar taught in textbooks the same as the English spoken on the street? No, English speakers ignore grammar all the time. But the differences aren't as major as you think. You should learn English that your textbooks teach and you should be able to understand most informal speech. If you don't understand something, just ask.