A non Muslim if i recall correctly asked me about Arabic, this is detailed response but bear in Mind I did not give the full reason as to why every Muslim should learn it, I simply gave a brief overview of it.
I would say it is definitely dependant on why you want to learn the language in the first place, if you want to learn it for the sake of tourism, or to visit an Arab country and experience different cultures then your study of the Arabic will not be a painstaking one, however if you wish to understand the true essence of the Arabic language, it's history, it's gestation, it's rise to the zenith of academia, and it's slow decline from a scholarly perspective then it will definitely have to be studied from an Islamic approach.
What you will find on this site is many people speaking to you about Arabic from a non scholarly angle, you will find people who will tell you to learn the language because they are born 'Arabs' who want you to come to their country and eat the food that they eat, or to take a ride on some camels in return for the much needed dollar or something of that sort.
What you will seldom find from those very people, is a true appreciation of the language and how it came to be the only living semitic language which continues to thrive unlike many semitic languages which were deemed dead for over a vast time in history. So what exactly made the Arabic language what became?
The Arabic language originated in southern Arabia in what was known and is still known as Yemen, the Yemenites spoke and formed the very foundation of the current language which is spoken today, as time passed, the Language encompassed the entire Arabian Peninsula. As these groups of people formed, like in any other human habitat, tribes were formed based on lineage and location, as these tribes all spoke one language there was a somewhat uniform way of speaking coupled with dialects used by each tribe when speaking amongst themselves. The Arabs by this time had started to write poetry, different topics, and different types, known as (النثر) Nathr and (القصيدة) Al Qaseedah. Some of the most prominent poems would be hung off walls and the most prominent of those were the 7 poems hung on the walls of the K'aba in Makkah known as (Al Mu'Allaqaatus Sab'ah) the Seven Pasted Poems. This bring us into the political and economic situation of the Arabs and their tongue during that particular time.
We are now entering the Pre Islamic faze of the Arabic language, it is no longer a language which is simple in vocabulary or weak in its grammatical structure, the Arabic language has entered into a phase just shy of being interred into the select few languages which can be considered great in the history of the world, such as Amharic, Roman, Greek, Sanskrit amongst many others, the Arabs find themselves in a arid harsh land wedged between the Romans to the north, the Ethiopians to west and the Persians to the east. At this time, there a words which are slowly entering the Arabic language through trade and assimilation. Like previously mentioned the Arabs are currently speaking the language in the purest form ever known, of course, loanwords find itself into the language through trade and travel, however the language at this point is Regarded amongst scholars of the Arabic Language as the purest form the language had ever witnessed.
Once at the peak, the only way is down, and it is well regarded in the Arab world that had this event not taken place, the Arabic language would have found itself in the graveyard with other dead Semitic languages, however something happens which changes the trajectory of the Arabic Language, and that is the Qur'an. I do not want to get into the whole topic but it is needless to say that every breathing piece of Arabic literature can be traced back to the Qur'an in some way shape or form. The Qur'an was the very first piece of Arabic written and documented into book form and produced on a mass scale, the language structure used in it, the phrases, the vocabulary and the rhetoric found within makes it the single most important documented piece of writing to land between the two hands of anyone wanting to embark on learning Arabic. From the moment of it first being recited by the Prophet Muhammad peace be upon him, the Arabic language becomes a source of literature and from it, the deluge of Arabic writings we have witnessed for the past 1400 years, but that still does not answer a few valuable questions.
Even at this time the language has not been codified manually ready to be taught to fresh converts of the new faith, it has not been given Technical Definitions to differentiate particle from verb, or noun or verb etc....however it is quite late and the list goes on. What i would say, is that is after the Qur'an is revealed a new wave of budding scholars and students seeking to understand the religion study the language and disciplines begin to form such as Nahw, (Grammar) however the early muslims would call it Al I'raab Al Binna, also Sarf (Morphology), coupled with the study of the Qur'an Tafseer and the science of the Prophetic Narration Ilmul Hadeeth the Arabic language enters a stage it has never known, because of the large expansion of Islam into North Africa, Egypt, And the East, many words begin to find itself into the language not previously seen on such a scale, causing many scholars to go into the Deserts of Arabia and live with the Bedouins, writing down every pure word of Arabic they know in order to preserve it at least for Academia, which benefitted us today, it is probably worth mentioning many of the greatest Scholars of this era were not Arab but had a pure love for the language because of the Qur'an, making the set out on sometimes very hostile and dangerous quests for knowledge.
You should definitely read the Qur'an after learning Arabic, you will be amazed with what you see, and we are forever indebted to what those early generations did for us in the path of truth and enlightenment.